Quantum Complexity Theory* 1 Introduction
Quantum complexity theory
y z
Ethan Bernstein Umesh Vazirani
September
Abstract
In this pap er we study quantum computation from a complexity theoretic viewp oint Our rst result
is the existence of an e cientuniversal quantum Turing Machine in Deutsch s mo del of a quantum Turing
Machine This construction is substantially more complicated than the corresp onding construction
for classical Turing Machines in fact even simple primitives such as lo oping branching and comp osition
are not straightforward in the context of quantum Turing Machines We establish how these familiar
primitives can b e implemented and also intro duce some new purely quantum mechanical primitives
suchaschanging the computational basis and carrying out an arbitrary unitary transformation of
p olynomially b ounded dimension
We also consider the precision to which the transition amplitudes of a quantum Turing Machine need
to b e sp eci ed We prove that O log T bits of precision su ce to supp ort a T step computation This
justi es the claim that that the quantum Turing Machine mo del should b e regarded as a discrete mo del
of computation and not an analog one
We give the rst formal evidence that quantum Turing Machines violate the mo dern complexity
theoretic formulation of the Church Turing thesis Weshow the existence of a problem relativeto
an oracle that can b e solved in p olynomial time on a quantum Turing Machine but requires sup er
p olynomial time on a b ounded error probabilistic Turing Machine and thus not in the class BPP
The class BQP of languages that are e ciently decidable with small error probability on a quantum
P
Turing Machine satis es BPP BQP P Therefore there is no p ossibility of giving a math
ematical pro of that quantum Turing Machines are more p owerful than classical probabilistic Turing
Machines in the unrelativized setting unless there is a ma jor breakthrough in complexity theory
Intro duction
Just as the theory of computability has its foundations in the Church Turing thesis computational com
plexity theory rests up on a mo dern strengthening of this thesis which asserts that any reasonable mo del
of computation can b e e ciently simulated on a probabilistic Turing Machine an e cient simulation is
one whose running time is b ounded by some p olynomial in the running time of the simulated machine
Here we take reasonable to mean in principle physically realizable Some mo dels of computation though
interesting for other reasons do not meet this criterion For example it is clear that computers that op er
ate on arbitrary length words in unit time or that exactly compute with in nite precision real numb ers are
A preliminary abstract of this work app eared as
y
Microsoft Corp oration One Microsoft Way Redmond WA ethanb microsoft com Research supp orted by N S F
Grant CCR
z
Computer Science Division University of California Berkeley CA vazirani cs berkeley edu Research sup
p orted by N S F Grant CCR
not realizable It has b een argued that the Turing Machine mo del actually the p olynomial time equiva
lent cellular automaton mo del is the inevitable choice once we assume that we can implement only nite
precision computational primitives Given the widespread b elief that NP BPP this would seem to put
a wide range of imp ortant computational problems the NP hard problems well b eyond the capabilityof
computers
However the Turing Machine fails to capture all physically realizable computing devices for a funda
mental reason the Turing Machine is based on a classical physics mo del of the Universe whereas current
physical theory asserts that the Universe is quantum physical Can we get inherently new kinds of discrete
computing devices based on quantum physics Early work on the computational p ossibilities of quantum
physics asked the opp osite question do es quantum mechanic s insistence on unitary evolution restrict
the class of e ciently computable problems They concluded that as far as deterministic computation is
concerned the only additional constraint imp osed by quantum mechanics is that the computation must b e
reversible and therefore by Bennett s work it follows that quantum computers are at least as p owerful as
classical computers The issue of the extra computational p ower of quantum mechanics over probabilistic
computers was rst raised bybyFeynman in In that pap er Feynman p ointed out a very curious
problem the natural simulation of a quantum physical system on a probabilistic Turing Machine requires
an exp onential slowdo wn Moreover it is unclear how to carry out the simulation more e ciently In
view of Feynman s observation wemust re examine the foundations of computational complexity theory
and the complexity theoretic form of the Church Turing thesis and study the computational p ower of
computing devices based on quantum physics
A precise mo del of a quantum physical computer hereafter referred to as the quantum Turing Machine
was formulated by Deutsch There are twoways of thinking ab out quantum computers One way
that may app eal to computer scientists is to think of a quantum Turing Machine as a quantum physical
analogue of a probabilistic Turing Machine it has an in nite tap e and a transition function and the
actions of the machine are lo cal and completely sp eci ed by this transition function Unlike probabilistic
Turing Machines quantum Turing Machines allow branching with complex probability amplitudes but
imp ose the further requirement that the machine s evolution b e time reversible This view is elab orated
in x Another way is to view a quantum computer as e ecting a transformation in a space of complex
sup erp ositions of con gurations Quantum physics requires that this transformation b e unitary A quan
tum algorithm may then b e regarded as the decomp osition of a unitary transformation into a pro duct of
unitary transformations each of whichmakes only simple lo cal changes This view is elab orated in x
Both formulations play an imp ortant role in the study of quantum computation
One imp ortant concern is whether quan tum Turing Machines are really analog devices since they
involve complex transition amplitudes It is instructive to examine the analogous question for probabilistic
Turing Machines There one mightworry that probabilistic machines are not discrete and therefore not
reasonable since they allow transition probabilities to b e real numbers However there is extensive
work showing that probabilistic computation can b e carried out in a suchaway that it is so insensitive
to the transition probabilities that they can b e allowed to vary arbitrarily in a large range In
this pap er we show in a similar sense that quantum Turing Machines are discrete devices the transition
amplitudes need only b e accurate to O log T bits of precision to supp ort T steps of computation As
Lipton p ointed out it is crucial that the numb er of bits is O log T and not O T as it was in an
early version of this pap er since k bits of precision requires pinning down the transition amplitude to one
k
part in Since the transition amplitude is some physical quantity such as the angle of a p olarizer or the
length of a pulse wemust not assume that we can sp ecify it to b etter than one part in some p olynomial
in T and therefore the precision must b e O log T
Another basic question one may ask is whether it is p ossible to de ne the notion of a general purp ose
quantum computer In the classical case this question is answered a rmatively byshowing that there is
an e cient universal Turing Machine In this pap er we prove that there is an e cient quantum Turing
Machine When given as input the sp eci cation of an arbitrary QTM M an input x to M a time b ound
T and an accuracy the universal machine pro duces a sup erp osition whose Euclidean distance from the
time T sup erp osition of M on x is at most Moreover the simulation time is b ounded by a p olynomial
in T jxj and Deutsch gaveadi erent construction of a universal quantum Turing Machine
The simulation overhead in Deutsch s construction is exp onential in T the issue Deutschwas interested
in was computability not computational complexity The structure of the e cient universal quantum
Turing Machine constructed in this pap er is very simple It is just a deterministic Turing Machine with
a single typ e of quantum op eration a quantum coin ip an op eration that p erforms a rotation on a
single bit The existence of this simple universal quantum Turing Machine has a b earing on the physical
realizability of quantum T uring Machines in general since it establishes that it is su cienttophysically
realize a simple quantum op eration on a single bit in addition to maintaining coherence and carrying out
deterministic op erations of course Adleman et al and Solovay and Yao have further clari ed
this p ointbyshowing that quantum coin ips with amplitudes and are su cient for universal
quantum computation
Quantum computation is necessarily time reversible since quantum physics requires unitary evolution
This makes it quite complicated to correctly implementeven simple primitives such as lo oping branching
and comp osition These are describ ed in x In addition we also require programming primitives such
as changing the computational basis which are purely quantum mechanical These are describ ed in x
Another imp ortant primitive is the ability to carry out any sp eci ed unitary transformation of p olynomial
dimension to a sp eci ed degree of accuracy Inx we showhow to build a quantum Turing that implements
this primitive Finally all these pieces are put together in x to construct the universal quantum Turing
Machine
We can still ask whether the quantum Turing Machine is the most general mo del for a computing device
based on quantum physics One approach to arguing a rmatively is to consider various other reasonable
mo dels and to show that the quantum Turing Machine can e ciently simulate each of them An earlier
version of this w ork left op en the question of whether standard variants of a quantum Turing Machine
such as machines with multiple tap es or with mo di ed tap e access are more p owerful than the basic mo del
Yao showed that these mo dels are p olynomially equivalent to the basic mo del as are quantum circuits
whichwere intro duced in The e ciency of Yao s simulation has b een improved in to show that
the simulation overhead is a p olynomial with degree indep endent of the numb er of tap es Arguably the
full computational p ower of quantum physics for discrete systems is captured by the quantum analog of
a cellular automaton It is still an op en question whether a quantum cellular automaton might b e more
powerful than a quantum Turing Machine there is also an issue ab out the correct de nition of a quantum
cellular automaton The di culty has to do with decomp osing a unitary transformation that represents
manyoverlapping sites of activityinto a pro duct of simple lo cal unitary transformations This problem
has b een solved in the sp ecial case of linearly b ounded quantum cellular automata
Finally several researchers have explored the computational p ower of quantum Turing Machines Early
work by Deutsch and Jozsa showed how to exploit some inherently quantum mechanical features
of QTMs Their results in conjunction with subsequent results by Berthiaume and Brassard
established the existence of oracles under which there are computational problems that QTMs can solvein
ty whereas if we require a classical probabilistic Turing machine to pro duce p olynomial time with certain
the correct answer with certainty then it must take exp onential time on some inputs On the other hand
these computational problems are in BPP the class of problems that can b e solved in p olynomial time
by probabilistic Turing machines that are allowed to give the wrong answer with small probability Since
BPP is widely considered the class of e ciently computable problems these results left op en the question
of whether quantum computers are more p owerful than classical computers
In this pap er wegive the rst formal evidence that quantum Turing Machines violate the mo dern form
of the Church Turing thesis byshowing that relative to an oracle there is a problem that can b e solved in
o log n
p olynomial time on a quantum Turing Machine but cannot b e solved in n time on a probabilistic
Turing Machine with any xed error probability A detailed discussion ab out the implications of
these oracle results is in the intro duction to x Simon subsequently strengthened our result in the
time parameter byproving the existence of an oracle relative to which a certain problem can b e solved
n
in p olynomial time on a quantum Turing Machine but cannot b e solved in less than steps on a
probabilistic Turing Machine Simon s problem is in NP co NP and therefore do es not address the
non determinism issue More imp ortantly Simon s pap er also intro duced an imp ortant new technique
whichwas one of the ingredients in a remarkable result proved subsequently by Shor Shor gave
p olynomial time quantum algorithms for the factoring and discrete log problems These two problems
have b een well studied and their presumed intractability forms the basis of much of mo dern cryptography
These results hav e injected a greater sense of urgency to the actual implementation of a quantum computer
The class BQP of languages that are e ciently decidable with small error probability on a quantum
P
Turing Machine satis es BPP BQP P This rules out the p ossibility of giving a mathematical
pro of that quantum Turing Machines are more p owerful than classical probabilistic Turing Machines in
the unrelativized setting unless there is a ma jor breakthrough in complexity theory
It is natural to ask whether quantum Turing Machines can solveevery problem in NP in p olynomial
time Bennett Bernstein Brassard and Vazirani give evidence showing the limitations of quantum
Turing Machines They show that relative to an oracle chosen uniformly at random with probability
n
the class NP cannot b e solved on a quantum Turing Machine in time o They also show that relative
to a p ermutation oracle chosen uniformly at random with probability the class NP co NP cannot
n
b e solved on a quantum Turing Machine in time o The former b ound is tight since recentwork
of Grover shows how to accept the class NP relativetoany oracle on a quantum computer in time
n
O
Several designs have b een prop osed for realizing quantum computers A numb er of authors
have argued that there are fundamental problems in building quantum computers most notably the e ects
of the decoherence of quantum sup erp ositions or the entanglement of the system with the environment
Very recently there have b een a sequence of imp ortant results showing how to implement quantum error
correcting co des and also use these co des to make quantum algorithms quite robust against the e ects
of decoherence and
Quantum computation touches up on the foundations of b oth computer science and quantum physics
The nature of quantum physics was clari ed by the Einstein Po dolsky Rosen paradox and Bell s inequalities
discussed in which demonstrate the di erence b etween its statistical prop erties and those of any
classical mo del The computational di erences b etween quantum and classical physics are if anything
more striking and can b e exp ected to o er new insights into the nature of quantum physics For example
one might naively argue that it is imp ossible to exp erimentally verify the exp onentially large size of
the Hilb ert space asso ciated with a discrete quantum system since any observation leads to a collapse
of its sup erp osition However an exp eriment demonstrating the exp onential sp eedup o ered by quantum
computation over classical computation would establish that something like the exp onentially large Hilb ert
space must exist Finally it is imp ortant as Feynman p ointed out to clarify the computational
overhead required to simulate a quantum mechanical system The simple form of the universal quantum
Turing Machine constructed here the fact that it has only a single non trivial quantum op eration de ned
on a single bit suggests that even very simple quantum mechanical systems are capable of universal
quantum computation and are therefore hard to simulate on classical computers
This pap er is organized as follows Section intro duces some of the mathematical machinery and
notation we will use In x weintro duce the quantum Turing Machine as a natural extension of classical
probabilistic Turing Machines We also show that quantum Turing Machines need not b e sp eci ed with
an unreasonable amount of precision In x and x we demonstrate the basic constructions which will
allow us to build up large complicated quantum Turing Machines in subsequent sections Many actions
which are quite easy for classical machines such as completing a partially sp eci ed machine running one
machine after another or rep eating the op eration of a machine a given numb er of times will require non
trivial constructions for quantum Turing Machines In x weshowhow to build a single quantum Turing
Machine which can carry out any unitary transformation which is provided as input Then in x weuse
this simulation of unitary transformations to build a universal quantum computer Finally inx we give
our results b oth p ositive and negative on the p ower of quantum Turing Machines
Preliminaries
Let C denote the eld of complex numb ers For C wedenoteby its complex conjugate
Let V be a vector space over C An inner pro duct over V is a complex function de ned on V V
which satis es
x V x x Moreover x x i x
x y z V x y z x z y z
x y V x y y x
The inner pro duct yields a norm given by kxk x x In addition to the triangle inequality
kx y k kxk ky k the norm also satis es the Schwarz inequality k x y k kxkky k
An inner product space is a vector space V together with an inner pro duct
An inner pro duct space H over C is a Hilbert space if it is complete under the induced norm where
H is complete if every Cauchy sequence converges i e if fx g is a sequence with x H such that
n n
lim kx x k then there is an x in H with lim kx xk
n m n m n n
Given any inner pro duct space V eachvector x V de nes a linear functional x V C where
x y x y The set of such linear functionals is also an inner pro duct space and will b e referred to as
the vector dual of V and denoted V In the case that V is a Hilb ert space V is called the dual of V and
is the set of all continuous linear functionals on V and the dual space V is also a Hilb ert space
In Dirac s notation a vector from an inner pro duct space V is identi ed using the ket notation ji
with some symb ol s placed inside to distinguish that vector from all others We denote elements of the
dual space using the bra notation hj Thus the dual of j i is h j and the inner pro duct of vectors j i
and j i which is the same as the result of applying functional h j to the vector j i is denoted by h j i
Let U b e a linear op erator on V In Dirac s notation we denote the result of applying U to j i as
U j i U also acts as a linear op erator on the dual space V mapping each linear functional h j of the dual
space to the linear functional which applies U followed by h j We denote the result of applying U to h j
by h jU
For any inner pro duct space V we can consider the usual vector space basis or Hamel basis fj ig
i i I
Every vector j i V can b e expressed as a nite linear combination of basis vectors In the case that
k k and h j j i for i j we refer to the basis as an orthonormal basis With resp ect to an
i i j
P
orthonormal basis we can write eachvector j i V as j i j i where h j i Similarly
i i i i
i I
P
each dual vector h j V can b e written as as h j h j where h j i Thus each element
i i i i
i I
j i V can b e thought of as a column vector of the s and each element h j V can b e thoughtof
i
as a rowvector of the s Similarly each linear op erator U may b e represented b y the set of matrix
i
elements fh jU j ig arranged in a square matrix with rows and columns b oth indexed by I Then
i j i j I
the column of U with index i is the vector U j i and the row of U with index i is the dual vector
i
h jU
i
For a Hilb ert space H fj ig is a Hilbert spacebasis for H if it is a maximal set of orthonormal
i i I
vectors in H Every vector j i H can b e expressed as the limit of a sequence of vectors each of which
can b e expressed as a nite linear combination of basis vectors
Given a linear op erator U in an inner pro duct space if there is a linear op erator U which satis es
hU j i h jU i for all then U is called the adjoint or Hermitian conjugate of U If a linear
op erator in an inner pro duct space has an adjoint it is unique The adjoint of a linear op erator in a
Hilb ert space or in a nite dimensional inner pro duct space always exists It is easy to see that if the
adjoints of U and U exist then U U U U and U U U U An op erator U is called
Hermitian or self adjoint if it is its own adjoint U U The linear op erator U is called unitary if its
adjoint exists and satis es U U UU I
If we represent linear op erators as square matrices indexed over an orthonormal basis then U is
represented by the conjugate transp ose of U So in Dirac s notation wehave the convenient identity
h jU j i h jU j i
Recall that if the inner pro duct space V is the tensor pro duct of two inner pro duct spaces V V then
for each pair of vectors j i V j i V there is an asso ciated tensor pro duct j i j i in V In
Dirac s notation we denote j i j i as j ij i
xik A linear op erator is called bounded if kU k is The norm of U is de ned as kU k sup kU j
kjxik
nite We will freely use the following standard facts ab out b ounded linear op erators
If U exists then kU k kU k
kU U k kU kkU k
kU k kU k kU U k kU k kU k
Notice that a unitary op erator U must satisfy kU k We will often use the following fact which
tells us that if we approximate a series of unitary transformations with other unitary transformations the
error increases only additively
Fact If U U U U are unitary transformations on an inner pro duct spacethen
kU U U U k kU U k kU U k
This fact follows from Statements and ab ove since
kU U U U k kU U U U k kU U U U k
kU U kkU k kU kkU U k
Miscellaneous notation
If d is a direction fL Rg then d is the opp osite of d
Given two probability distributions P and P over the same domain I the total variation distance
P
between P and P is equal to jP i P i j
i I
We will refer to the cardinality ofasetS as card S and the length of a string x as jxj
Quantum Turing Machines
Aphysics like view of randomized computation
Before we formally de ne a quantum Turing Machine QTM weintro duce the necessary terminology in
the familiar setting of probabilistic computation As a b onus we will b e able to precisely lo cate the p oint
of departure in the de nition of a QTM
Quantum mechanics makes a distinction b etween a system s evolution and its measurement In the
absence of measurement the time evolution of a probabilistic TM can b e describ ed by a sequence of
probability distributions The distribution at eachstepgives the likeliho o d of each p ossible con guration
of the machine We can also think of the probabilistic TM as sp ecifying an in nite dimensional sto chastic
matrix M whose rows and columns are indexed by con gurations Each column of this matrix gives
the distribution resulting from the corresp onding con guration after a single step of the machine If we
represent the probability distribution at one time step byavector jv i then the distribution at the next
step is given by the pro duct M jv i Inquantum physics terminology we call the distribution at each step a
linear sup erp osition of con gurations and we call the co e cientofeach con guration its probability
its amplitude The sto chastic matrix is referred to as the time evolution op erator
Three comments are in order First not every sto chastic matrix has an asso ciated probabilistic TM
Sto chastic matrices obtained from probabilistic TM are nitely sp eci ed and map each con guration by
making only lo cal changes to it Second the supp ort of the sup erp osition can b e exp onential in the run
ning time of the machine Third we need to constrain the entries allowed in the transition function of our
probabilistic TM Otherwise it is p ossible to smuggle hard to compute quantities into the transition ampli
th
tudes for instance by letting the i bit indicate whether the ith deterministic TM halts on a blank tap e
g More generally we might allow A common restriction is to allow amplitudes only from the set f
any real numb er in the interval which can b e computed by some deterministic algorithm to within
n
any desired in time p olynomial in n Itiseasilyshown that the rst p ossibility is computationally no
more restrictive than the second
Returning to the evolution of the probabilistic TM when we observe the machine after some number
of steps we do not see the linear sup erp osition probability distribution but just a sample from it
If we observe the entire machine then we see a con guration sampled at random according to the
sup erp osition The same holds if we observe just a part of the machine In this case the sup erp osition
collapses to one that corresp onds to the probability distribution conditioned on the value observed By
the linearity of the law of alternatives the mere act of making observations at times earlier than t do es
not change the probability for each outcome in an observation at time t So even though the unobserved
sup erp osition mayhave supp ort that grows exp onentially with running time we need only keep trackof
a constant amount of information when simulating a probabilistic TM which is observed at eachstep
The computational p ossibilities of quantum physics arise out of the fact that observing a quantum system
changes its later b ehavior
Recall that a matrix is sto chastic if it has non negativerealentries that sum to in each column
The law of alternatives says exactly that the probabilityofanevent A do esn t change if we rstcheck to see whether
event B has happ ened P A P AjB P B P Aj B P B
De ning a QTM
Our mo del of randomized computation is already surprisingly close to Deutsch s mo del of a QTM The
ma jor change that is required is that in quantum physics the amplitudes in a system s linear sup erp osition
and the matrix elements in a system s time evolution op erator are allowed to b e complex numb ers rather
than just p ositive reals When an observation is made the probability asso ciated with each con guration is
not the con guration s amplitude in the sup erp osition but rather the squared magnitude of its amplitude
So instead of always having a linear sup erp osition whose entries sum to we will nowalways have a linear
sup erp osition whose Euclidean length is This means that QTMs must b e de ned so that their time
evolution preserves the Euclidean length of sup erp ositions
Making these changes to our mo del wearrive at the following de nitions
For completeness let us recall the de nition of a deterministic TM There are many standard variations
to the de nition of deterministic TM none of which a ect their computational p ower In this pap er we will
make our choices consistent with those in Deutsch s pap er we consider TMs with a two way in nite
tap e and a single tap e head whichmust move left or right one square on each step We also give standard
de nitions for interpreting the input output and running time of a deterministic TM Note that although
we usually restrict our discussion to TMs with tap e head movements fL Rg we will sometimes consider
generalized TMs with tap e head movements fL N Rg where N means no head movement
De nition A deterministic T uring Machine is de ned by a triplet Q where is a nite
alphabet with an identi ed blank symbol Q is a nite set of states with an identi ed initial state q and
nal state q q and the deterministic transition function is a function
f
Q Q fL Rg
The TM has a two way in nite tapeofcel ls indexedby Z and a single read write tapehead that moves
along the tape
A con guration or instantaneous description oftheTMisacomplete description of the the contents of
the tape the location of the tapehead and the state q Q of the nite control At any time only a nite
number of tapecel ls may contain non blank symbols
For any con guration c of TM M the successor con guration c is de ne d by applying the transition
function to the current state q and currently scannedsymbol in the obvious way We write c c to
M
denote that c fol lows from c in one step
By convention we requiretheinitial con guration of M to be satisfy the fol lowing conditions the tape
head is in cel l cal led the start cell and the machine is in state q An initial con guration has input
x if x is written on the tapeinpositions and al l other tapecel ls are blank The TM
halts on input x if it eventual ly enters the nal state q The number of steps a TM takes to halt on input
f
x is its running time on input x If a TM halts then its output is the string in consisting of those tape
contents from the leftmost non blank symbol to the rightmost non blank symbol or the empty string if the
eforecomputes a function from to entire tape is blank A TM which halts on al l inputs ther
Wenow give a slightly mo di ed version of the de nition of a QTM provided byDeutsch As in
the case of probabilistic TM wemust limit the transition amplitudes to e ciently computable numb ers
Adleman et al and Solovay and Yao have separately shown that further restricting QTMs to
rational amplitudes do es not reduce their computational p ower In fact they have shown that the set
of amplitudes f g are su cient to construct a universal QTM Wegive a de nition of the
computation of a QTM with a particular string as input but we defer discussing what it means for a QTM
to halt or give output until x Again we will usually restrict our discussion to QTMs with tap e head
movements fL Rg but will sometimes consider generalized QTMs with tap e head movements fL N Rg
As we p ointed out in the intro duction unlike in the case of deterministic TMs these choices do makea
greater di erence in the case of QTMs This p oint is also discussed later in the pap er
De nition Cal l C the set consisting of C such that there is a deterministic algorithm that
n
computes the real and imaginary parts of to within in time polynomial in n
A quantum Turing Machine QTM M is de nedbyatriplet Q where is a nite alphabet
with an identi ed blank symbol Q is a nite set of states with an identi ed initial state q and nal
state q q and the quantum transition function is a function
f
Q fL Rg
Q C
The QTM has a two way in nite tapeofcel ls indexedbyZ and a single read write tapehead that moves
along the tape We de ne con gurations initial con gurations and nal con gurations exactly as for
deterministic TMs
Let S be the inner product space of nite complex linear combinations of con gurations of M with the
Euclidean norm We cal l each element S a sup erp osition of M The QTM M de nes a linear operator
U S S cal led the time evolution op erator of M as fol lows If M starts in con guration c with current
M
P
state p and scannedsymbol Then after one step M wil l beinsuperposition of con gurations c
i i
i
whereeach non zero corresponds to a transition p q d and c is the new con guration that results
i i
from applying this transition to c Extending this map to the entirespace S through linearity gives the
linear time evolution operator U
M
Note that we de ned S by giving an orthonormal basis for it namely the con gurations of M In terms
of this standard basis each sup erp osition S may b e represented as a vector of complex numb ers indexed
by con gurations The time evolution op erator U may b e represented by the countable dimensional
M
square matrix with columns and rows indexed by con gurations where the matrix element from column
c and row c gives the amplitude with which con guration c leads to con guration c in a single step of M
For convenience we will overload notation and use the expression p q d to denote the amplitude
in p of j ijq ijdi
The next de nition provides an extremely imp ortant condition that QTMs must satisfy to b e consistent
with quantum physics Wehaveintro duced this condition in the form stated b elow for exp ository purp oses
As we shall see later in x there are other equivalent formulations of this condition that are more familiar
to quantum physics
De nition We wil l say that M is well formed if the its time evolution operator U preserves
M
Euclidean length
Well formedness is a necessary condition for a QTM to b e consistent with quantum physics As we shall
see in the next subsection well formedness is equivalent to unitary time evolution which is a fundamental
requirement of quantum physics
Next we de ne the rules for observing the QTM M For those familiar with quantum mechanics we
should state that the de nition b elow restricts the measurements to b e in the computational basis of S
This is b ecause the actual basis in which the measurement is p erformed must b e e ciently computable
and therefore wemay without loss of generality p erform the rotation to that basis during the computation
itself
P
De nition When QTM M in superposition c is observedormeasured con guration c
i i i
i
is seen with probability j j Moreover the superposition of M is updatedto c
i
We may also perform a partial measurement say only on the rst cel l of the tape In this case sup
P
pose that the rst cel l may contain the values or and suppose the superposition was c
i i
i
P
c where the c are those con gurations that have a in the rst cel l and c are those con g
i i i i
i
P
in the rst cel l Measuring the rst cel l results in Pr j j Moreover urations that have a
i
i
P
p
if a is observed the new superposition is given by c i e thepart of the superposition
i i
i
Pr
consistent with the answer with amplitudes scaledtogiveaunitvector
Note that the well formedness condition on a QTM simply says that the the time evolution op erator
of a QTM must satisfy the condition that in each successive sup erp osition the sum of the probabilities of
all p ossible con gurations must b e
Notice that a QTM di ers from a classical TM in that the user has decisions b eyond just cho osing an
input A priori it is not clear whether multiple observations might increase the p ower of QTMs This p oint
is discussed in more detail in and there it is shown that one may assume without loss of generality
that the QTM is only observed once Therefore in this pap er we shall make simplifying assumptions
ab out the measurement of the nal result of the QTM The fact that these assumptions do not result in
any loss of generality follows from the results in
In general the output of a QTM is a sample from a probability distribution We can regard two
QTMs as functionally equivalent for practical purp oses if their output distributions are su ciently close
to each other A formal de nition of what it means for one QTM to simulate another is also given in
As in the case of classical TMs the formal de nition is quite unwieldy In the actual constructions it will
b e easy to see in what sense they are simulations Therefore we will not replicate the formal de nitions
from here Wegive a more informal de nition b elow
De nition We say that QTM M simulates M with slowdown f with accuracy if the fol lowing
holds let D be a distribution such that observing M on input x after T steps produces a sample from D
let D be a distribution such that observing M on input x after f T steps produces a sample from D
Then we say that M simulates M with accuracy if jD D j
We will sometimes nd it convenient to measure the accuracy of a simulation by calculating the Eu
clidean distance b etween the target sup erp osition and the sup erp osition achieved by the simulation The
following shows that the variation distance b etween the resulting distributions is at most times this
Euclidean distance
Lemma Let S such that k k k k and k k Then the total variation distance
is at most between the probability distributions resulting from measurements of and
P P
Pro of Let jii and jii Observing gives each jii with probability j j while
i i i
i i
P
observing gives each jii with probability j j Let jii Then the latter
i i i
i
probability j j can b e expressed as
i
j j j j
i i i i i i i i i
i i i
Therefore the total variation distance b etween these two distributions is at most
X X
j j j h jj i h jj i k kk k j j k k j
i i i i
i i
i i
e Finally note that since wehave unit sup erp ositions wemust hav
Quantum computing as a unitary transformation
In the preceding sections weintro duced QTMs as an extension of the notion of probabilistic TMs We
stated there that a QTM is well formed if it preserves the norm of the sup erp ositions In this section we
explore a di erent and extremely useful alternative view of QTMs in terms of prop erties of the time
evolution op erator We prove that a QTM is well formed i its time evolution is unitary Indeed unitary
time evolution is a fundamental constraint imp osed by quantum mechanics and wechose to state the
well formedness condition in the last section mainly for exp ository purp oses
Understanding unitary evolution from an intuitivepoint of view is quite imp ortant to comprehending
the computational p ossibilitie s of quantum mechanics Let us explore this in the setting of a quantum
mechanical system that consists of n parts each of whichcanbeinoneoftwo states lab eled j i and j i
these could b e n particles each with a spin state If this were a classical system then at any given instant
it would b e in a single con guration which could b e describ ed by n bits However in quantum physics the
system is allowed to b e in a linear sup erp osition of con gurations and indeed the instantaneous state of
n
the system is describ ed by a unit vector in the dimensional vector space whose basis vectors corresp ond
n
to all the con gurations Therefore to describ e the instantaneous state of the system wemust sp ecify
n
complex numb ers The implications of this are quite extraordinary even for a small system consisting
of particles nature must keep trackof complex numb ers just to rememb er its instantaneous
state Moreover it must up date these numb ers at each instanttoevolve the system in time This is an
extravagant amount of e ort since is larger than the standard estimates on the numb er of particles in
the visible universe So if nature puts in suchextravagantamounts of e ort to evolveeven a tiny system
at the level of quantum mechanics it would make sense that we should design our computers to take
advantage of this
However unitary evolution and the rules for measurement in quantum mechanics place signi cant
constraints on how these features can b e exploited for computational purp oses One of the basic primitives
that allows these features to b e exploited while resp ecting the unitarity constraints is the discrete fourier
transform this is describ ed in more detail in x Here we consider some very simple cases One
interesting phenomenon supp orted by unitary evolution is the interference of computational paths In
a probabilistic computation the probabilityofmoving from one con guration to another is the sum of
the probabilities of each p ossible path from the former to the latter The same is true of the probability
amplitudes in quantum computation but not necessarily of the probabilities of observations Consider for
p p
example applying the transformation U twice in sequence to the same tap e cell which
p p
at rst contains the symbol If we observe the cell after the rst application of U we see either symbol
with probability If we then observe after applying U a second time the symb ols are again equally
likely However since U is the identity ifwe only observe at the end we see a with probability
So even though there are two computational paths that lead from the initial to a nal symb ol they
interfere destructively cancelling each other out The two paths leading to a interfere constructively so
that even though b oth have probability when we observetwice wehave probability of reaching if
we observe only once Such a b o osting of probabilities by an exp onential factor lies at the heart of the
QTM s advantage over a probabilistic TM in solving the Fourier sampling problem
Another constraint inherent in computation using unitary transformations is reversibility Weshowin
R
x that for anyQTM M there is a corresp onding QTM M whose time evolution op erator is the con
jugate transp ose of the time evolution op erator of M and therefore undo es the actions of M Sections
and are devoted to de ning the machinery to deal with this feature of QTMs
We prove b elow in App endix A that a QTM is well formed if and only if its time evolution op erator
is unitary This establishes that our de nition which did not mention unitary evolution for exp ository
purp oses do es satisfy the fundamental constraint imp osed by quantum mechanics unitary evolution
Actually one could still ask why this is consistent with quantum mechanics since the space S of nite
linear combinations of con gurations is not a Hilb ert space since it is not complete To see that this
do esn t present a problem notice that S is a dense subset of the Hilb ert space H of all not just nite
linear combinations of con gurations Moreover any unitary op erator U on S has a unique extension U
to H and U is unitary and its inverse is U The pro of is quite simple Let U and U b e the continuous
extensions of U and U to H Letx y H Then there are sequences fx g fy g S such that x x
n n n
and y y Moreover for all n Ux y x U y Taking limits we get that Ux y x U y
n n n n n
as desired
As an aside we should brie y mention that another resolution of this issue is achieved by following
Feynman who suggested that if we use a quantum mechanical system with Hamiltonian U U then
the resulting system has a lo cal time invariant Hamiltonian It is easy to probabilistically recover the
computation of the original system from the computation of the new one
It is interesting to note that the following theorem would not b e true if we de ned QTMs using a
one way in nite tap e In that case the trivial QTM which always moves its tap e head rightwould b e
well formed but its time evolution would not b e unitary since it s start con guration could not b e reached
from any other con guration
Theorem A A QTM is wel l formed i its time evolution operator is unitary
Precision required in a QTM
One imp ortant concern is whether QTMs are really analog devices since they involve complex transition
amplitudes The issue here is how accurately these transition amplitudes must b e sp eci ed to ensure that
the correctness of the computation is not compromised In an earlier version of this pap er we showed that
O
T bits of precision are su cient to correctly carry out T steps of computation to within accuracy for
Lipton pointed out that for the device to b e regarded as a discrete device wemust any constant
O
require that its transition amplitudes b e sp eci ed to at most one part in T as opp osed to accurate to
O
within T bits of precision This is b ecause the transition amplitude represents some physical quantity
such as the angle of a p olarizer or the length of a pulse and wemust not assume that we can sp ecify
it to b etter than one part in some p olynomial in T and therefore the numb er of bits of precision must b e
O log T This is exactly what weproved shortly after Lipton s observation and we present that pro of in
this section
The next theorem shows that b ecause of the unitary time evolution errors in the sup erp osition made
over a sequence of steps will in the worst case only add
Theorem Let U be the time evolution operator of a QTM M and T Ifj i j i j i j i
T T
are superpositions of U such that
kj i j ik
i i
j i U j i
i i
T
then kj i U j ik T
T
Pro of Let j i j i j i Thenwehave
i i i
T T T
j i U j i U j i U j i j i
T T
The theorem follows by the triangle inequalitysinceU is unitary and kj ik
i
De nition We say that QTMs M and M are close if they have the same state set and alphabet
and if the di erencebetween each pair of corresponding transition amplitudes has magnitude at most
Note that M and M can be close even if one or both are not wel l formed
The following theorem shows that two QTMs which are close in the ab ove sense give rise to time
evolution op erators which are close to each other even if the QTMs are not well formed As a simple
consequence the time evolution op erator of a QTM is always b ounded even if the QTM is not well
formed
Theorem If QTMs M and M with alphabet and state set Q are close then the di erencein
their time evolutions has norm at most card card Q Moreover this statement holds even if one or
both of the machines are not wel l formed
Pro of Let QTMs M and M with alphab et and state set Q be given whichare close Let U be the
time evolution of M and let U b e the time evolution of M
P
Now consider any unit length sup erp osition of con gurations j i jc i Thenwe can express
j j
j
the di erence in the machines op eration on j i as follows
X X
A
U j i U j i jc i
i j i j
i j
j
i P j
isthesetof i such that con guration c can lead to c in a single step of M or M and where where P j
i j
and are the amplitudes with which c leads to c in M and M
i j i j
i j
Applying the triangle inequality and the fact that the square of the sum of n reals is at most n times
the sum of their squares wehave
P P
kU j i U j ik
i j i
j i P j
i j
P P
card card Q
i j i
i P j j
i j
Then since M and M are close wehave
P P
card card Q
i i j
j i P j
i j
P P
j j card card Q
i i j
j i P j
i j
P P
card card Q j j
i
j i P j
Finally since for any con guration c there are at most card card Q con gurations that can lead
j
to c in a single step wehave
j
P P P
card card Q j j card card Q j j
i i
j i P j i
card card Q
Therefore for any unit length sup erp osition j i
k U U j ik card card Q
The following corollary shows that O log T bits of precision are su cient in the transition amplitudes
to simulate T steps of a QTM to within accuracy for any constant
Corollary Let M Q be a wel l formed QTM and let M be a QTM which is
card card Q T
close to M where ThenM simulates M for time T with accuracy Moreover this statment holds
even if M is not wel l formed
Pro of Let b Without loss of generality we further assume
card card Q T
Consider running M and M with the same initial sup erp osition Since M is well formed by Theo
rem A its time evolution op erator U is unitary By Theorem on page the time evolution
of U op erator of M U is within
T
Applying U can always b e expressed as applying U and then adding a p erturbation of length at most
times the length of the current sup erp osition So the length of the sup erp osition of U at time t is at
t
most Since T this length is at most e Therefore app ealing to Theorem ab ove the
di erence b etween the sup erp ositions of M and M at time T is a sup erp osition of norm at most T
Finally Lemma on page tells us that observing M at time T gives a sample from a distribution
which is within total variation distance of the distributions sampled from by observing M at time T
Input output conventions for QTMs
Timing is of crucial imp ortance to the op eration of a QTM b ecause computational paths can only interfere
if they take the same numb er of time steps Equally imp ortant are the p osition of the tap e head and
alignment of the tap e contents In this subsection weintro duce several input output conventions on
QTMs and deterministic TMs which will help us maintain these relationships while manipulating and
combining machines
Wewould like to think of our QTMs as nishing their computation when they reach the nal state
q However it is unclear howwe should regard a machine whichreaches a sup erp osition in whichsome
f
con gurations are in state q but others are not We try to avoid such di culties bysaying that a QTM
f
halts on a particular input if it reaches a sup erp osition consisting entirely of con gurations in state q
f
De nition A nal con guration of a QTM is any con guration in state q IfwhenQTMM is
f
run with input x at time T the superp osition contains only nal con gurations and at any time less than
T the superposition contains no nal con guration then M halts with running time T on input x The
superposition of M at time T is cal ledthe nal sup erp osition of M run on input x Apolynomial time
QTM is a wel l formed QTM which on every input x halts in time polynomial in the length of x
Wewould like to de ne the output of a QTM which halts as the sup erp osition of the tap e contents of
the con gurations in the machine s nal sup erp osition However wemust b e careful to note the p osition of
the tap e head and the alignment relative to the start cell in each con guration since these details determine
whether later paths interfere Recall that the output string of a nal con guration of a TM is its tap e
contents from the leftmost non blank symb ol to the rightmost non blank symb ol This means that giving
an output string leaves unsp eci ed the alignment of this string on the tap e and the lo cation of the tap e
head to b e identi ed When describing the input output b ehavior of a QTM we will sometimes describ e
this additional information When we do not the additional information will b e clear from context For
example we will often build machines in which all nal con gurations have the output string b eginning in
the start cell with the tap e head scanning its rst symbol
De nition A QTM is cal led well b ehaved if it halts on al l input strings in a nal superposition
whereeach con guration has the tapehead in the same cel l If this cel l is always the start cel l we cal l the
machine stationary Similarly a deterministic TM is cal led stationary if it halts on al l inputs with its tape
head back in the start cel l
To simplify our constructions we will often build QTMs and then combine them in simple ways like
running one after the other or iterating one a varying numb er of times Todosowemust b e able to add
new transitions into the initial state q of a machine However since there may already b e transitions
into q the resulting machine may not b e reversible But we can certainly redirect the transitions out
of the nal state q of a reversible TM or a well b ehaved QTM without a ecting its b ehavior Note that
f
for a well formed QTM if q alwa ys leads backtoq then there can b e no more transitions into q In
f
that case redirecting the transitions out of q will allow us to add new ones into q without violating
f
reversibility Wewillsay that a machine with this prop ertyisin normal form Note that a well b ehaved
QTM in normal form always halts b efore using any transition out of q and therefore also b efore using
f
any transition into q This means that altering these transitions will not alter the relevantpartofthe
machine s computation For simplicity we arbitrarily de ne normal form QTMs to step right and leave
the tap e unchanged as they go from state q to q
f
De nition A QTM or deterministic TM M Q is in normal form if
q j ijq ijRi
f
We will need to consider QTMs with the sp ecial prop erty that any particular state can b e entered while
the machine s tap e head steps in only one direction Though not all QTMs are unidirectional we will
sho w that any QTM can b e e ciently simulated by one that is Unidirectionality will b e a critical concept
in reversing a QTM in completing a partially describ ed QTM and in building our universal QTM We
further describ e the advantages of unidirectional machines after Theorem in x
De nition A QTM M Q is cal led unidirectional if each state can be enteredfrom only
one direction In other words if p q d and p q d areboth non zero then d d
Finally we will nd it convenient to use the common to ol of thinking of the tap e of a QTM or
deterministic TM as consisting of several tracks
De nition A multi track TM with k tracks is a TM whose alphabet is of the form
k
with a sp ecial blank symbol in each so that the blank in is Wespecify the input by
i
specifying the string on each track and optional ly by specifying the alignment of the contents of the
k
is started in the superposition consisting tracks So a TM run on input x x x
i k
i
th
only of the initial con guration with the non blank portion of the i coordinate of the tapecontaining
the string x starting in the start cel l More general ly on input x jy x jy x jy with x y the
i k k i i
i
th
non blank p ortion of the i track is x y aligned so that the rst symbol of each y is in the start cel l Also
i i i
input x x x with x x is abbreviatedasx x x
k l k l
Programming a QTM
In this section we explore the fundamentals of building up a QTM from several simpler QTMs Im
plementing basic programming primitives such as lo oping branching and reversing a computation is
straightforward for deterministic TMs However these constructions are more di cult for QTMs b ecause
one must b e very careful to maintain reversibility In fact the same di culties arise when building re
versible deterministic TMs However building reversible TMs up out of simpler reversible TMs has never
b een necessary This is b ecause Bennett showed how to e ciently simulate any deterministic TM with
one which is reversible So one can build a reversible TM by rst building the desired computation with
a non reversible machine and then using Bennett s construction None of the constructions in this section
makeany sp ecial use of the quantum nature of QTMs and in fact all techniques used are the same as
those required to make the analogous construction for reversible TMs
We will show in this section that reversible TMs are a sp ecial case of QTMs So as Deutsch noted
Bennett s result allows any desired deterministic computation to b e carried out on a QTM However
Bennett s result is not su cienttoallow us to use deterministic computations when building up QTMs
b ecause the di erent computation paths of a QTM will only interfere prop erly provided that they take
exactly the same numb er of steps We will therefore carry out a mo di ed version of Bennett s construction
to show that any deterministic computation can b e carried out byareversible TM whose running time
dep ends only on the length of its input Then di erent computation paths of a QTM will take the same
numb er of steps provided that they carry out the same deterministic computation on inputs of the same
length
Reversible Turing Machines
De nition A reversible Turing Machine is a deterministic TM for which each con guration has at
most one predecessor
Note that wehave altered the de nition of a reversible TM from the one used by Bennett sothat
our reversible TMs are a sp ecial case of our QTMs First wehave restricted our reversible TM to move
its head only left and right instead of also allowing it to stay still Second we insist that the transition
function b e a complete function rather than a partial function Finally we consider only reversible TMs
with a single tap e though Bennett worked with multi tap e machines
Theorem Any reversible TM is also a wel l formed QTM
Pro of The transition function of a deterministic TM maps the current state and symb ol to an up date
triple If we think of it as instead giving the unit sup erp osition with amplitude for that triple and
elsewhere then is also a quantum transition function and wehave a QTM The time evolution matrix
corresp onding to this QTM contains only the entries and Since Corollary B proven b elowin
App endix B tells us that each con guration of a reversible TM has exactly one predecessor this matrix
must b e a p ermutation matrix If the TM is reversible then there must b e at most one in eachrow
P
jc i is mapp ed by this time evolution matrix to some Therefore any sup erp osition of con gurations
i i
i
P
other sup erp osition of con gurations jc i So the time evolution preserves length and the QTM is
i
i i
well formed
Previous work of Bennett shows that reversible machines can e ciently simulate deterministic TMs
Of course if a deterministic TM computes a function which is not one to one then no reversible machine
can simulate it exactly Bennett showed that a generalized reversible TM can do the next b est thing
which istotakeany input x and compute x M x where M x is the output of M on input x He also
showed that if a deterministic TM computes a function which is one to one then there is a generalized
reversible TM that computes the same function For b oth constructions he used a multi tap e TM and
also suggested how the simulation could b e carried out using only a single tap e machine Morita et al
use Bennett s ideas and some further techniques to show that any deterministic TM can b e simulated by
a generalized reversible TM with a twosymb ol alphab et
We will give a slightly di erentsimulation of a deterministic TM with a reversible machine that
preserves an imp ortant timing prop erty
First we describ e why timing is of critical imp ortance In later sections we will build QTMs with
interesting and complex interference patterns However two computational paths can only interfere if they
reach the same con guration at the same time We will often want paths to interfere which run muchof
the same computation but with di erent inputs We can only b e sure they interfere if weknow that these
computations can b e carried out in exactly the same running time We therefore want to show that any
function computable in deterministic p olynomial time can b e computed by a p olynomial time reversible
TM in suchaway that the running time of the latter is determined entirely by the length of its input
Then provided that all computation paths carry out the same deterministic algorithms on the inputs of
the same length they will all take exactly the same numb er of steps
We prove the following theorem in App endix B on page using ideas from the constructions of Bennett
and Morita et al
Theorem Synchronization Theorem If f is a function mapping strings to strings which can
becomputed in deterministic polynomial time and such that the length of f x depends only on the length
of x then thereisapolynomial time stationary normal form reversible TM which given input x produces
output x f x and whose running time depends only on the length of x
If f is a function from strings to strings that such that both f and f can becomputed in deterministic
polynomial time and such that the length of f x depends only on the length of x then thereisapolynomial
time stationary normal form reversible TM which given input x produces output f x and whose running
time depends only on the length of x
Programming primitives
Wenow showhow to carry out several programming primitives reversibly The Branching Reversal and
Lo oping Lemmas will b e used frequently in subsequent sections
The pro ofs of the following two lemmas are straightforward and are omitted However they will b e
quite useful as we build complicated machines since they allow us to build a series of simpler machines
while ignoring the contents of tracks not currently b eing used
Lemma Given any QTM reversible TM M Q and any set thereisaQTM reversible
TM M Q such that M behaves exactly as M while leaving its second track unchanged
Lemma Given any QTM reversible TM M Q and permutation k
k
k there is a QTM reversible TM M Q suchthattheM behaves exactly
k
as M except that its tracks arepermutedaccording to
The following two lemmas are also straightforward but stating them separately makes Lemma
below easy to prove The rst deals with swapping transitions of states in a QTM We can swap the
outgoing transitions of states p and p for transition function by de ning p p p
p and p p for p p p Similarly wecanswap the incoming transitions of states q
and q by de ning p q d p q d p q d p q d and p q d
p q d for q q q
Lemma If M is a wel l formedQTM reversible TM then swapping the incoming or outgoing
transitions between a pairofstatesinM gives another wel l formedQTM reversible TM
Lemma Let M Q and M Q be two wel l formed QTMs reversible TMs with
the same alphabet and disjoint state sets Then then the union of the two machines M Q Q
and with arbitrarily chosen start state q Q Q is also a wel l formedQTM reversible TM
Using the two preceding lemmas we can insert one machine in place of a state in another When
we p erform such an insertion the computations of the t womachines might disturb each other However
sometimes this can easily b e seen not to b e the case For example in the insertion used by the Dovetailing
Lemma b elow we will insert one machine for the nal state of a well b ehaved QTM so that the computation
of the original machine has completed b efore the inserted machine b egins In the rest of the insertions we
carry out the twomachines will op erate on di erenttracks so the only p ossible disturbance involves the
p osition of the tap e head
Lemma If M and M are normal form QTMs reversible TMs with the same alphabet and q is
a state of M then thereisanormalformQTMM which acts as M except that each time it would enter
state q it instead runs machine M
Pro of Let M and M b e as stated with initial and nal states q q q q and with q a state of
f f
M
Then we can construct the desired machine M as follows First take the union of M and M according
to Lemma on page and make the start state q if q q and q otherwise and make the nal
state q if q q and q otherwise Then swap the incoming transitions of q and q and the outgoing
f f f
transitions of q and q according to Lemma on page to get the well formed machine M
f
Since M is in normal form the nal state of M leads back to its initial state no matter whether q is
the initial state of M the nal state of M or neither
Next we showhowtotaketwomachines and form a third by dovetailing one onto the end of the
other Notice that when wedovetail QTMs the second QTM will b e started with the nal sup erp osition of
the rst machine as its input sup erp osition If the second machine has di ering running times for various
t not halt even though the two original strings in this sup erp osition then the dovetailed machine migh
machines were well b ehaved Therefore a QTM built bydovetailing twowell b ehaved QTMs may not
itself b e well b ehaved
Lemma Dovetailing Lemma If M and M are wel l behaved normal form QTMs reversible
TMs with the same alphabet then there is a normal form QTM reversible TM M which carries out the
computation of M fol lowedbythecomputation of M
Pro of Let M and M be well b ehaved normal form QTMs reversible TMs with the same alphab et
and with start states and nal states q q q q
f f
To construct M we simply insert M for the nal state of M using Lemma on page
To complete the pro of we need to show that M carries out the computation of M followed by that
of M
To see this rst recall that since M and M are in normal form the only transitions into q and q
are from q and q resp ectively This means that no transitions in M havebeenchanged except for
f f
those into or out of state q Therefore since M is well b ehav ed and do es not prematurely enter state
f
q the machine M when started in state q will compute exactly as M until M would halt At that
f
point M will instead reach a sup erp osition with all con gurations in state q Then since no transitions
in M havebeenchanged except for those into or out of q M will pro ceed exactly as if M had b een
f
started in the sup erp osition of outputs computed by M
Now we showhow to build a conditional branch around two existing QTMs or reversible TMs The
branching machine will run one of the twomachines on its rst track input dep ending on its second track
input Since a TM can have only one nal state wemust rejoin the two branches at the end We can join
reversibly if we write back out the bit that determined whichmachine was used The construction will
simply build a reversible TM that accomplishes the desired branching and rejoining and then insert the
twomachines for states in this branching machine
Lemma Branching Lemma If M and M are wel l behaved normal form QTMs reversible
TMs with the same alphabet then there is a wel l behaved normal form QTM reversible TM M such
that if the second track is empty M runs M on its rst track and leaves its second track empty and if
the second track has a in the start cel l and al l other cel ls blank M runs M on its rst track and
leaves the where its tapehead ends up In either case M takes exactly four time steps morethanthe
appropriate M
i
Pro of Let M and M be well b ehaved normal form QTMs reversible TMs with the same alphab et
Then we can construct the desired QTM as follows We will showhow to build a stationary normal
form reversible TM BR whichalways takes four time steps and leaves its input unchanged always has a
with the following prop erties sup erp osition consisting of a single con guration and has two states q and q
If BR is run with a in the start cell and blanks elsewhere then BR visits q once with a blank tap e
and with its tap e head in the start cell and do esn t visit q at all and similarly if BR is run with a blank
tap e then BR visits q once with a blank tap e and with its tap e head in the start cell and do esn t visit q
at all Then if we extend M and M to have a second track with the alphab et of BR extend BR to have
a rst track with the common alphab et of M and M and insert M for state q and M for state q in
BR we will have the desired QTM M
We complete the construction by exhibiting the reversible TM BR The machine enters state q or q
dep ending on whether the start cell contains a and steps left and enters the corresp onding q or q while
stepping back right Then it enters state q while stepping left and state q while stepping back right
f
So we let BR have alphab et f g state set fq q q q q q q g and transition function de ned
f
by the following table
q q L q L
q q R
q R q
q q L
q q L
q q R
f
q q R q R
f
It can b e veri ed that the transition function of BR is one to one and that it can enter each state while
moving in only one direction Therefore app ealing to Theorem B on page it can b e completed to
give a reversible TM
Finally weshowhowtotake a unidirectional QTM or reversible TM and build its reverse Two
computational paths of a QTM will interfere only if they reach con gurations that do not di er in any
way This means that wemust b e careful when building a QTM to erase any information that might di er
along paths whichwewanttointerfere We will therefore sometimes use this lemma when constructing a
QTM to allow us to completely erase an intermediate computation
Since the time evolution of a well formed QTM is unitary we could reverse a QTM by applying the
unitary inverse of its time evolution However this transformation is not the time evolution of a QTM
since it acts on a con guration bychanging tap e symb ols to the left and right of the tap e head However
since each state in a unidirectional QTM can b e entered while moving in only one direction we can reverse
the head movements one step ahead of the reversal of the tap e and state Then the reversal will have its
tap e head in the prop er cell each time a symbol must b e changed
have the same alphabet then we say that M reverses the compu De nition If QTMs M and M
tation of M if identifying the nal state of M with the initial state of M and the initial state of M with
the nal state of M gives the fol lowing For any input x on which M halts if c and are the initial
x x
con guration and nal superposition of M on input x thenM halts on initial superposition with nal
x
superposition consisting entirely of con guration c
x
Lemma Reversal Lemma If M is a normal form reversible TM or unidirectional QTM then
there is a normal form reversible TM or QTM M that reverses the computation of M while taking two
extra time steps
Pro of Wewillprove the lemma for normal form unidirectional QTMs but the same argumentproves
the lemma for normal form reversible TMs
Let M Q b e a normal form unidirectional QTM with initial and nal states q and q and for
f
each q Q let d b e the direction which M must move while entering state q Then we de ne a bijection
q
on the set of con gurations of M as follows For con guration c in state q let c b e the con guration
derived from c bymoving the tap e head one square in direction d the opp osite of direction d Since
q q
the set of sup erp ositions of the form jci give an orthonormal basis for the space of sup erp ositions of M
we can extend to act as a linear op erator on this space of sup erp ositions by de ning jci j ci Itis
easy to see that is a unitary transformation on the space of sup erp ositions on M
Our new machine M will have the same alphab et as M and state set given by Q together with new
wing three statements su ce to prove that M reverses the initial and nal states q and q The follo
f
computation of M while taking two extra time steps
If c is a nal con guration of M and c is the con guration c with state q replaced by state q then
f
a single step of M takes sup erp osition jc i to sup erp osition jci
If a single step of M takes sup erp osition j i to sup erp osition j i where j i has no supp ort on
a con guration in state q then a single step of M takes sup erp osition j i to sup erp osition
j i
If c is an initial con guration of M and c is the con guration c with state q replaced by state q
f
then a single step of M takes sup erp osition j ci to sup erp osition jc i
To see this let x b e an input on which M halts and let j i j i b e the sequence of sup erp ositions
n
of M on input x so that j i jc i where c is the initial sup erp osition of M on x and j i has supp ort
x x n
only on nal con gurations of M Then since the time evolution op erator of M is linear Statement
tells us that if we form the initial con guration j i of M by replacing each state q in the j i with
f n
n
state q then M takes j i to j i in a single step Since M is in normal form none of j i j i
n n
n
have supp ort on any sup erp osition in state q Therefore Statement tells us that the next n steps of M
lead to sup erp ositions j i j i Finally Statement tells us that a single step of M maps
n
sup erp osition jc i to sup erp osition jc i
x
x
We de ne the transition function to giveawell formed M satisfying these three statements with
the following transition rules
q j ijq ijd i
f q
f
For each q Q q and each
X
q p q d j ijpijd i
q p
p
q j ijq ijd i
q
f
ijRi q j ijq
f
The rst and third rules can easily b e seen to ensure Statements and The second rule can b e
seen to ensure Statement as follows Since M is in normal form it maps a sup erp osition j i to a
sup erp osition j i with no supp ort on any con guration in state q if and only if j i has no supp ort
on any con gurations in state q Therefore the time evolution of M de nes a unitary transformation
f
from the space S of sup erp ositions of con gurations in states from the set Q Q q to the space of
f
sup erp ositions S of con gurations in states from the set Q Q q This fact also tells us that the
second rule ab ove de nes a linear transformation from space S back to space S Moreover if M takes
con guration c with a state from Q leads with amplitude to con guration c with a state from Q
then M takes con guration c to con guration c with amplitude Therefore the time evolution
of M on space S is the comp osition of and its inverse around the conjugate transp ose of the time
evolution of M Since this conjugate transp ose must also b e unitary the second rule ab ove actually gives
a unitary transformation from the space S to the space S which satis es Statement ab ove
Since M is clearly in normal form we complete the pro of byshowing that M is well formed Tosee
this just note that each of the four rules de nes a unitary transformation to one of a set of four mutually
orthogonal subspaces of the spaces of sup erp ositions of M
The Synchronization Theorem will allowustotake an existing QTM and put it inside a lo op so that
the machine can b e run any sp eci ed numb er of times
Building a reversible machine that lo ops inde nitely is trivial However if wewant to lo op some nite
numb er of times we need to carefully construct a reversible entrance and exit As with the Branching
Lemma ab ove the construction will pro ceed by building a reversible TM that accomplishes the desired
lo oping and then inserting the QTM for a particular state in this lo oping machine However there are
several di culties First in this construction as opp osed to the branching construction the reversible TM
leaves an intermediate computation written on its tap e while the QTM runs This means that inserting a
non stationary QTM would destroy the prop er functioning of the reversible TM Second even if we insert a
stationary QTM the second and any subsequent time the QTM is run it may b e started in sup erp osition
on inputs of di erent lengths and hence may not halt There is therefore no general statementwe are able
to makeaboutthebehavior of the machine once the insertion is carried out Instead we describ e here the
reversible lo oping TM constructed in App endix C on page and argue ab out sp eci c QTMs resulting
from this machine when they are constructed
Lemma Lo oping Lemma There is a stationary normal form reversible TM M andaconstant
c with the fol lowing properties On input any positive integer k written in binary M runs for time
c
O k log k and halts with its tape unchanged Moreover M has a special state q such that on input k M
visits state q exactly k times each time with its tapehead back in the start cel l
Changing the basis of a QTM s computation
In this section weintro duce a fundamentally quantum mechanical feature of quantum computation namely
changing the computational basis during the evolution of a QTM In particular we will nd it useful to
change to a new orthonormal basis for the transition function in the middle of the computation each
state in the new state set is a linear combination of states from the original machine
This will allowustosimulate a general QTM with one that is unidirectional It will also allowusto
prove that any partially de ned quantum transition function which preserves length can b e extended to
giveawell formed QTM
We start by showing howtochange the basis of the states of a QTM Then wegive a set of conditions
for a quantum transition function that are necessary and su cient for a QTM to b e well formed The last
of these conditions called separability will allow us to construct a basis of the states of a QTM which will
allow us to prove the Unidirection and Completion Lemmas b elow
The change of basis
If we takeawell formed QTM and cho ose a new orthonormal basis for the space of sup erp ositions of its
states then translating its transition function into this new basis will give another well formed QTM that
evolves just as the rst under the change of basis Note that in this construction the new QTM has the
same time evolution op erator as the original machine However the states of the new machine di er from
those of the old This change of basis will allowustoprove Lemmas and b elow
Q
Lemma Given a QTM M Q andasetofvectors B from C which forms an orthonormal
Q
basis for C thereisaQTMM B which evolves exactly as M under a change of basis from Q
to B
Q
Pro of Let M Q b e a QTM and B an orthonormal basis for C
Since B is an orthonormal basis this establishes a unitary transformation from the space of sup erp o
sitions of states in Q to the space of sup erp ositions of states in B Sp eci call y for each p Q wehavethe
mapping
X
hpjjv ijv i jpi
v B
Similarly wehave a unitary transformation from the space of sup erp ositions of con gurations with states
in Q to the space of con gurations with states in B In this second transformation a con guration with
state p is mapp ed to the sup erp osition of con gurations where the corresp onding con guration with state
v app ears with amplitude hpjjv i
Let us see what the time evolution of M should lo ok like under this change of basis In M a con guration
in state p reading a evolves in a single time step to the sup erp osition of con gurations corresp onding to
the sup erp osition p
X
p p q d j ij q ijdi
q d
With the change of basis the sup erp osition on the right hand side will instead b e
X X
hq jv i p q d j ijv ijdi
q
v d
Now since the state symb ol pair jv ij i in M corresp onds to the sup erp osition
X
hv jpijpij i
p
in M we should havein M
X X X
A
v hv jpi hq jv i p q d j ijv ijdi
p q
v d
Therefore M will b ehave exactly as M under the change of basis if we de ne bysaying that for
each v B
X X
v hv jpihq jv i p q d j ijv ijdi
p q
v d
Q
Since the vectors in B are contained in C each amplitude of is contained in C
Finally note that the time evolution of M must preserve Euclidean length since it is exactly the time
evolution of the well formed M under the change of basis
Lo cal conditions for QTM well formedness
In our discussion of reversible TMs in App endix B we nd prop erties of a deterministic transition function
which are necessary and su cient for it to b e reversible Similarly our next theorem gives three prop erties
of a quantum transition function which together are necessary and su cient for it to b e well formed The
rst prop erty insures that the time evolution op erator preserves length when applied to any particular
con guration Adding the second insures that the time evolution preserves the length of sup erp ositions
of con gurations with their tap e head in the same cell The third prop erty which concerns restricted
up date sup erp ositions handles sup erp ositions of con gurations with tap e heads in di ering lo cations This
third prop erty will b e of critical use in the constructions of Lemmas and b elow
De nition We wil l also refer to the superposition of states
X
p q d jq i
q Q
resulting from restricting p to those pieces writing symbol and moving in direction d as a direction
d going restricted sup erp osition denotedby p j d
Theorem A QTM M Q is wel l formed i the fol lowing conditions hold
Unit length
p Q k p k
Orthogonality
p p Q p p
Separability
p p Q p j L p j R
Pro of Let U b e the time evolution of a prop osed QTM M Q Weknow M is well formed
i U exists and U U gives the identity or equivalently i the columns of U have unit length and are
mutually orthogonal Clearly the rst condition sp eci es exactly that each column has unit length In
general con gurations whose tap es di er in a cell not under either of their heads or whose tap e heads
o cells apart cannot yield the same con guration in a single are not either in the same cell or exactly tw
step Therefore such pairs of columns are guaranteed to b e orthogonal and we need only consider pairs of
con gurations for which this is not the case The second condition sp eci es the orthogonality of pairs of
columns for con gurations that di er only in that one is in state p reading while the other is in state
p reading
Finally wemust consider pairs of con gurations with their tap e heads two cells apart Such pairs can
only interfere in a single step if they di er at most in their states and in the symb ol written in these cells
The third condition sp eci es the orthogonality of pairs of columns for con gurations which are identical
except that the second has its tap e head two cells to the left is in state p instead of p has a instead
of a in the cell under its tap e head and has a instead of a two cells to the left
Now consider again unidirectional QTMs those in whicheach state can b e entered while moving in
only one direction When we considered this prop erty for deterministic TMs it meant that when lo oking
at a deterministic transition function we could ignore the direction up date and think of as giving a
bijection from the current state and tap e symb ol to the new symb ol and state Here if is a unidirectional
quantum transition function then it certainly satis es the separability condition since no left going and
right going restricted sup erp ositions have a state in common Moreover up date triples will always share
the same direction if they share the same state Therefore a unidirectional is well formed i ignoring the
direction up date gives a unitary transformation from sup erp ositions of curren t state and tap e symbol
to sup erp ositions of new symb ol and state
Unidirection and Completion Lemmas
The separability condition of Theorem allows us to simulate any QTM with a unidirectional QTM by
applying a change of basis The same change of basis will also allow us to complete any partially sp eci ed
quantum transition function which preserves length
It is straightforward to simulate a deterministic TM with one which is unidirectional Simply split each
state q into two states q and q b oth of whicharegiven the same transitions that q had and then edit
r l
the transition function so that transitions moving rightinto q enter q and transitions moving left into q
r
enter q The resulting machine is clearly not reversible since the transition function op erates the same on
l
each pair of states q q
r l
To simplify the unidirection construction we rst showhowtointerleave a series of quantum transition
functions
Lemma Given k state sets Q Q Q Q and k transition functions each mapping from
k k
one state set to the next
Q fL Rg
i
Q C
i i
S
such that each preserves length there is a wel l formedQTMM with state set Q i which alternates
i i
i
stepping according to eachofthe k transition functions
Pro of Supp ose wehave k state sets transition functions as stated ab ove Then welet M b e the QTM
S
Q i and with with the same alphab et with state set given by the union of the individual state sets
i
i
transition function according to the
i
X
p i p q d j ijq i ijdi
i
q d
Clearly the machine M alternates stepping according to It is also easy to see that the time
k
evolution of M preserves length If j i is a sup erp osition of con gurations with squared length in the
i
subspace with con gurations with states from Q then maps to a sup erp osition with squared length
i
in the subspace with con gurations with states from Q
i i
Lemma Unidirection Lemma Any QTM M is simulated with slowdown by a factor of by
a unidirectional QTM M Furthermore if M is wel l behaved and in normal form then so is M
Pro of The key idea is that the separability condition of a well formed QTM allows a change of basis to
a state set in whicheachstatecanbeentered from only one direction
The separability condition says that
p p Q
p j R p j L
Q
This means that we can split C into mutually orthogonal subspaces C and C such that span C C
L R L R
Q
C and
p Q
p j d C
d
Now as shown in Lemma ab ove under a change of basis from state set Q to state set B the new
transition function is de ned by
X
v v d hv jpihq jv i p q d
p q
So cho ose orthonormal bases B and B for the spaces C and C and let M B B be
L R L R L R
the QTM constructed according to Lemma whichevolves exactly as M under a change of basis from
state set Q to state set B B B Then any state in M can b e entered in only one direction Tosee
L R
P
this rst note that since p j d B and v hv jq ijq i the separability condition implies that for
d
q
v B
d
X
p q d hv jq i
q
Therefore for any v v B B
d
X
v v d hv jpihq jv i p q d
p q
X X
hv jpi hq jv i p q d
p q
Therefore any state in B can b e entered while traveling in only one direction
Unfortunately this new QTM M might not b e able to simulate M The problem is that the start
state and nal state of M might corresp ond under the change of basis isomorphism to sup erp ositions of
states in M meaning that wewould b e unable to de ne the necessary start and nal states for M To x
this problem we use ve time steps to simulate each step of M and interleave the ve transition functions
using Lemma on page pagerefinterleaving lemma
Step rightleaving the tap e and state unchanged
p j ijpijRi
Change basis from Q to B while stepping left
X
p hpjbij ijbijLi
b B
M carries out a step of the computation of M So is just the quantum transition function
from QTM M constructed ab ove
Change basis back from B to Q while stepping left
X
b hbjpij ijpijLi
p Q
Step rightleaving the tap e and state unchanged
p j ijpijRi
If we construct QTM M with state set Q f g B f g using Lemma on page and
let M have start and nal states q and q then M simulates M with slowdown by a factor of
f
Next wemust show that each of the ve transition functions ob eys the well formedness conditions and
hence according to Lemma that the interleaved machine is well formed
The transition function certainly ob eys the well formedness conditions since M is a well
formed QTM Also and ob ey the three well formedness conditions since they are deterministic and
reversible Finally the transition functions and satisfy the unit length and orthogonality conditions
since they implementachange of basis and they ob ey the separability condition since they only movein
one direction
Finally wemust show that if M is w ell b ehaved and in normal form then we can make M well b ehaved
and in normal form
So supp ose M is well b ehaved and in normal form Then there is a T such that at time T the
sup erp osition includes only con gurations in state q with the tap e head back in the start cell and at any
f
time less than T the sup erp osition contains no con guration in state q But this means that when M
f
is run on input x the sup erp osition at time T includes only con gurations in state q with the tap e
f
head back in the start cell and the sup erp osition at any time less than T contains no con guration in
state q Therefore M is also well b ehaved
f
Then for any input x there is a T such that M enters a series of T sup erp ositions of con gurations
all with states in Q fq q g and then enters a sup erp osition of con gurations all in state q with the
f f
tap e head back in the start cell Therefore on input xM enters a series of T sup erp ositions of
con gurations all with states in Q f q q g and then enters a sup erp osition of con gurations all
f
in state q with the tap e head back in the start cell Therefore M is well b ehaved Also swapping the
f
outgoing transitions of q and q which puts M in normal form will not change the computation
f
of M on any input x
When we construct a QTM we will often only b e concerned with a subset of its transitions Luckily
any partially de ned transition function that preserves length can b e extended to giveawell formed QTM
De nition AQTMM whose quantum transition function is only de ned for a subset S Q
is cal leda partial QTM If the de ned entries of satisfy the threeconditions of Theorem on page
then M is cal ledawell formed partial QTM
Suppose M is a wel l formedpartial QTM with quantum transi Lemma Completion Lemma
tion function Then there is a wel l formedQTMM with the same state set and alphabet whose quantum
transition function agrees with wherever the latter is de ned
Pro of We noted ab ove that a unidirectional quantum transition function is well formed i ignoring the
direction up date it gives a unitary transformation from sup erp ositions of current state and tap e symbol
to sup erp ositions of new symb ol and state So if our partial QTM is unidirectional then we can easily ll
in the unde ned entries of by extending the set of up date sup erp ositions of to an orthonormal basis
for the space of sup erp ositions of new symb ol and state
For a general we can use the technique of Lemma on page to change the basis of away from
Q so that eachstatecanbeentered while moving in only one direction extend the transition function
and then rotate back to the basis Q
We can formalize this as follows
Let M Q bea well formed partial QTM with de ned on the subset S Q Denote by
S the complementofS in Q
Q
As in the pro of of the Unidirection Lemma ab ove the separability condition allows us to partition C
Q
into mutually orthogonal subspaces C and C such that span C C C and
L R L R
p S
p j d C
d
Then wecho ose orthonormal bases B and B for C and C and consider the unitary transformation
L R L R
from sup erp ositions of con gurations with states in Q to the space of con gurations with states in B
B B where any con guration with state p is mapp ed to the sup erp osition of con gurations where the
L R
corresp onding con guration with state v app ears with amplitude hpjv i
If we call the partial function followed by this unitary change of basis then wehave
X X
p hq jv i p q d j ijv ijdi
q
v d
Since preserves length and is followed by a unitary transformation also preserves length
But now can enter any state in B while moving in only one direction To see this rst note that
P
since p d B and v hv jq ijq i the separability condition implies that for v B
d
q
d
X
p q d hv jq i
q
Therefore for any p v S B
d
X
p v d hq jv i p q
q
Therefore any state in B can b e entered while traveling in only one direction
Then since the direction is implied by the new state we can think of as mapping the current state
and symb ol to a sup erp osition of new symb ol and state Since preserves length the set S isa setof
orthonormal vectors and we can expand this set to an orthonormal basis of the space of sup erp ositions
of new symb ol and state Adding the appropriate direction up dates and assigning these new vectors arbi
trarily to S wehave a completely de ned that preserves length Therefore assigning S as S
followed by the inverse of the basis transformation gives a completion for that preserves length
An e cient QTM implementing anygiven unitary transformation
Supp ose that the tap e head of a QTM is con ned to a region consisting of k contiguous tap e cells the tap e
is blank outside this region Then the time evolution of the QTM can b e describ ed byad dimensional
k
unitary transformation where d k card Q card In this section we showconversely that there is a
QTM that given any d dimensional unitary transformation as input carries out that transformation on a
region of its tap e Tomake this claim precise wemust sayhow the d dimensional unitary transformation
is sp eci ed We assume that an approximation to the unitary transformation is sp eci ed bya d d complex
matrix whose entries are approximations to the entries of the actual unitary matrix corresp onding to the
desired transformation Weshow in Theorem on page that there is a QTM which on input and
p
a d dimensional transformation which is within distance of a unitary transformation carries out a
d
d
transformation whichisan approximation to the desired unitary transformation Moreover the running
time of the QTM is b ounded by a p olynomial in d and
In a single step a QTM can map a single con guration into a sup erp osition of a b ounded number of
con gurations Therefore in order to carry out an approximation to an arbitrary unitary transformation
on a QTM we showhow to approximate it by a pro duct of simple unitary transformations each such
simple transformation acts as the identity in all but two dimensions We then show that there is a particular
simple unitary transformation such that anygiven simple transformation can b e expressed as a pro duct of
p ermutation matrices and p owers of this xed simple matrix Finally we put it all together and showhow
to design a single QTM that carries out an arbitrary unitary transformation this QTM is deterministic
except for a single kind of quantum coin ip
The decomp osition of an arbitrary unitary transformation into a pro duct of simple unitary transforma
tions is similar to work carried out by Deutsch Deutsch s work although phrased in terms of quantum
computation networks can b e viewed as showing that a d dimensional unitary transformation can b e de
comp osed into a pro duct of transformations where each applies a particular unitary transformation to
dimensions and acts as the identity elsewhere Wemust consider here several issues of e ciency not ad
dressed by Deutsch First we are concerned that the decomp osition contain a numb er of transformations
which is p olynomial in the dimension of the unitary transformation and in the desired accuracy Second
we desire that the decomp osition can itself b e e ciently computed given the desired unitary transforma
tion as input For more recentwork on the e cientsimulation of a unitary transformation by a quantum
computation network see and the references therein
Measuring errors in approximated transformations
In this section we will deal with op erators linear transformations on nite dimensional Hilb ert spaces
It is often convenient to x an orthonormal basis for the Hilb ert space and describ e the op erator bya
nite matrix with resp ect to the chosen basis Let e e b e an orthonormal basis for Hilb ert space
d
d th
H C Then we can represent an op erator U on H byad d complex matrix M whose i j entry
T th
M and we will denote it by M We denote m is Ue e The i row of the matrix M is given by e
i i j j i i
th
by M the conjugate transp ose of M The j column of M is given by Me U the adjointofU is
i i j
represented by the d d matrix M M is unitary i MM M M I It follows that if M is unitary
then the rows and columns of M are orthonormal
Recall that for a b ounded linear op erator U on a Hilb ert space H the norm of U is de ned as
jU j sup jUxj
kxk
If we represent U by the matrix M thenwe can de ne the norm of the matrix M to b e same as the norm
of U Thus since we re working in a nite dimensional space
kM k max kMvk
kv k
Fact If U is unitary then kU k kU k
Pro of x H kUxk Ux Ux x U Ux x x kxk Therefore kU k and similar
reasoning shows kU k
We will nd it useful to keep trackofhow far our approximations are from b eing unitary We will use
the following simple measure of a transformation s distance from b eing unitary
De nition Abounded linear operator U is cal led close to unitary if there is a unitary operator U
such that kU U k Ifwerepresent U by the matrix M thenwecan equivalently say that M is close
to unitary if there is a unitary matrix M such that kM M k
Notice that app ealing to Statement in x if U is close to unitary then kU k However
the converse is not true For example the linear transformation has norm but is away from
unitary
Next we show that if M is close to unitary then the rows of M are close to unit length and will b e
close to orthogonal
Lemma If a d dimensional complex matrix M is close to unitary then
kM k
i
i j kM M k
i j
Pro of Let M b e the unitary matrix such that kM M k LetN M M Then we know that for
each i kM k and kN k
i i
So for any i wehave M N M Therefore kM k
i i i i
Next since M is unitary itfollows that for i j M M Therefore M N M N
i i i
j j j
Expanding this as a sum of four terms weget
N M M M N N M N
i i i i
j j j j
Since kM k and kN k the Schwarz inequality tells us that kM N k kN M k
i i
j j
and kN N k Using the triangle inequalitywe conclude that kM M k Therefore
i i
j j
k kM M
i j
We will also use the following standard fact that a matrix with small entries must have small norm
Lemma If M is a d dimensional squarecomplex matrix such that jm j for al l i j then
i j
kM k d
Pro of If eachentry of M has magnitude at most then clearly eachrow M of M must have norm at
i
p
most d So if v is a d dimensional column vector with jv j wemust have
X X
kMvk kM v k d d
i
i i
where the inequality follows from the Scwarz Inequality Therefore kM k d
Decomp osing a unitary transformation
Wenow describ e a class of exceedingly simple unitary transformations whichwe will b e able to carry out
using a single QTM These near trivial transformations either apply a phase shift in one dimension or
apply a rotation b etween two dimensions while acting as the identity otherwise
De nition A d d unitary matrix M is near trivial if it satis es one of the fol lowing two conditions
i
M is the identity except that one of its diagonal entries is e for some i e jm
j j
i
e k jm and k lm
k k k l
M is the identity except that the submatrix in one pair of distinct dimensions j and k is the rotation by
cos sin
some angle So as a transformation M is near trivial if there exists
sin cos
and i j such that Me cos e sin e Me sin e cos e and k i j M e e
i i j j i j k k
We callatransformation which satis es the former a near trivial phase shift andwecal l a transformation
which satis es the latter a near trivial rotation
i
We will write a near trivial matrix M in the following way IfM is a phase shift of e in dimension
j then we will write down j j and if M is a rotation of angle between dimensions j and k we will
write down j k This convention guarantees that the matrix that we are sp ecifying is a near trivial
matrix and therefore a unitary matrix even if for precision reasons we write down an approximation to
the matrix that we really wish to sp ecify This feature will substantially simplify our error analyses
Before we sho whow to use near trivial transformations to carry out an arbitrary unitary transformation
we rst showhow to use them to map any particular vector to a desired target direction
d
Lemma There is a deterministic algorithm which on input a vector v C andabound
computes near trivial matrices U U such that
d
kU U v kv ke k
d
where e is the unit vector in the rst coordinate direction The running time of the algorithm is bounded
by a polynomial in d log and the length of the input
d
Pro of First we use d phase shifts to map v into the space IR We therefore want to apply to each
v
i
dimension i with v the phase shift So weletP b e the near trivial matrix which applies to
i i
kv k
i
Re v
i
dimension i the phase shift by angle where ifv and otherwise cos
i i i i
kv k
i
Re v
i
th
dep ending whether Im v is p ositive or negative Then P P v is the vector with i or cos
i d
kv k
i
co ordinate kv k
i
Next weuse d rotations to move all of the weightofthevector into dimension So we let R be
i
the near trivial matrix which applies to dimensions i and i the rotation by angle where
i
kv k
i
q
cos
i
P
d
kv k
j
j i
if the sum in the denominator is not and otherwise Then
i
R R P P v kv ke
d d
Now instead of pro ducing these values exactly we can in time p olynomial in d and log and
i i
the length of the input compute values and which are within of the desired values Call
i i
d kv k
P and R the near trivial matrices corresp onding to P and R but using these approximations Then
i i
i i
since the distance b etween p oints at angle and on the unit circle in the real plane is at most j j
kR R k
i
i
Thinking of the same inequality on the unit circle in the complex plane wehave
kP P k
i
i
Finally since each matrix P P R R is unitary Fact on page gives us
i i
i i
R P P k d kR R P P R
d d
d d
and therefore
kR R P P v kv ke k d kv k
d d
Wenow showhow the ability to map a particular vector to a desired target direction allows us to
approximate an arbitrary unitary transformation
Theorem There is a deterministic algorithm running in time polynomial in d and log and the
length of the input which when given as input U where and U is a d d complex matrix which is
p
close to unitary computes d dimensional near trivial matrices U U with n polynomial in d
n
d
d
such that kU U U k
n
Pro of First weintro duce notation to simplify the pro of Let U be a d d complex matrix Then wesay
U is k simple if its rst k rows and columns are the same as those of the d dimensional identity Notice
that the pro duct of two d dk simple matrices is also k simple
If U were d simple wewould have U I and the desired computation would b e trivial In general
the U which our algorithm must approximate will not even b e simple So our algorithm will pro ceed
th
through d phases such that during the i phase the remaining problem is reduced to approximating a
matrix whichisi simple
ximate a k simple U with a series of near trivial matrices with the pro duct Supp ose we start to appro
V Then to approximate U wewould still need to pro duce a series of near trivial matrices whose pro duct
W satis es W UV To reduce the problem wemust therefore compute near trivial matrices whose
pro duct V is such that UV is close to b eing k simple We can accomplish this by using the algorithm
of Lemma ab ove
So let U b e given whichisk simple and is close to unitary and let Z be the lower right d k d k
T
submatrix of Z Weinvoke the pro cedure of Lemma on inputs Z the vector corresp onding to the
rst rowofZ and The output is a sequence of d k dimensional near trivial matrices V V
d k
T
has the prop erty that kVZ kZ ke k such that their pro duct V V V
d k
Now supp ose that we extend V and the V backtod dimensional k simple matrices and we let
i
W UV Then clearly V is unitary and V and W are k simple In fact since V is unitary and U is
close to unitary W is also close to unitary Moreover W is close to b eing k simple as desired We
st st T
k and that the entries of the k will show b elow that the k rowofW satis es kW e
k
k
column of W satisfy kw k for j k
j k
So let X b e the d d k simple matrix such that
x x for j x for j x w for j l l
j j j l j l
It follows from our b ounds on the norm of the rst rowof W and on the entries of the rst column of W
p p
that kW X k d Since W is close to unitary we can then conclude that X is d close
to unitary
Unfortunately we cannot compute the entries of W UV exactly Instead app ealing to Lemma
on page we compute them to within d to obtain a matrix W such that kW W k Let s use
the entries of W to de ne matrix X analogous to X Using the triangle inequality it is easy to see
p p
that kW X k d and X is d close to unitary Ifwe are willing to incur an error of
p
kW X k d then we are left with the problem of approximating the k simple X by a pro duct
of near trivial matrices Therefore wehave reduced the problem of approximating the k simple matrix
U by near trivial matrices to the problem of approximating the k simple matrix X by near trivial
matrices while incurring two sources of error
p
d since we are approximating X instead of W An error of kW X k
p
The new matrix X is only d close to unitary
p
d Clearly is an upp er b ound on b oth the sources of error cited ab ove Therefore Let
p p
P
d
j d
the total error in the approximation is just d d The last inequality follows since
j
p
d and therefore the sum can b e b ounded by a geometric series Therefore the total error in the
p
approximation is b ounded by since by assumption U is close to unitary for
d
d
It is easy to see that this algorithm runs in time p olynomial in d and log Our algorithm consists of
d iterations of rst calling the algorithm from Lemma on page to compute V and then computing
p
d
d
the matrix X Since the each iteration takes time p olynomial in d and log these d calls take a total
time p olynomial in d and log
st T
Finally we show as required that the k rowofW satis es kW e k and that the
k
k
st
entries of the k column of W satisfy kw k for j k To see this rst recall that
j k
T
the lower dimension V satis es kVZ kZ ke k where Z is the rst rowofthelower right k k
T
k Then since submatrix of U Therefore the higher dimension V satis es kVU kU ke
k k
k
T st
kU k it follows that kVU e k Therefore the k rowof W satis es
k k
k
T
kW e k
k
k
st
Next we will show that this implies that the entries of the k column of W satisfy kw k
j k
for j k To see this rst notice that since V is unitary and U is delta close to unitary W is also
Now close to unitary This means that by Statement of Lemma on page W W
k
j
T
k This implies that jw j Also let us denote by let us use the condition kW e
k k k
k
W the d dimensional rowvector arrived at by dropping w from W Then the condition that W
j j k j k
T
also implies that kW k Also the fact that W is close to unitary implies that is close to e
k
k
kW k Putting all this together wehave W W w w W W
j k k k k j
j
j k
W W w w W W Therefore w w
k j k k k j k k
j k j k
Finally since wemay assume that wehave jw j Therefore jw j
j k j k
Carrying out near trivial transformations
In this section weshowhow to construct a single QTM that can carry out at least approximately any
sp eci ed near trivial transformation Since a near trivial transformation can apply an arbitrary rotation
either b etween two dimensions or in the phase of a single dimension wemust rst showhow a xed
rotation can b e used to e ciently approximate an arbitrary rotation Note that a single copy of this xed
rotation gives the only non classical amplitudes those other than in the transition function of the
universal QTM constructed b elow See Adleman et al and Solovay and Yao for the constructions
of universal QTMs whose amplitudes are restricted to a small set of rationals
P
i
Lemma Let R
i
Then there is a deterministic algorithm taking time polynomial in log and the length of the input
which on input with and produces integer output k boundedbya polynomial in
such that
jk R j mod
Pro of First we describ e a pro cedure for computing suchak
Next approximate Start by calculating n apower of such that as a fraction with
n
n n
denominator In other words nd an integer m such that
m
n n
n
Then we can let k m b ecause
X
i
n n
mo d m Rmo d m
i
X
i
n
A
mo d m
i log n
X
m i
n
A
m mo d
n
i log n
and since
X
i
n n n
m m
i log n
n n
n
wehave
m m
n n
jm R j mo d m R mo d
n n
n n
n
At this p oint it should b e clear that a single QTM can carry out any sequence of near trivial trans
formations to any desired accuracy We formalize this notion b elow byshowing that there is a QTM
that accepts as input the descriptions of a sequence of near trivial transformations and an error b ound
and applies an approximation of the pro duct of these transformations on anygiven sup erp osition The
formalization is quite tedious and the reader is encouraged to skip the rest of the subsection if the ab ove
statement is convincing
Below wegive a formal de nition of what it means for a QTM to carry out a transformation
De nition Let be the alphabet of the rst track of QTM M Let V bethecomplex vector
spaceofsuperpositions of k length strings over Let U bea linear transformation on V and let x
U
be a string that encodes U perhaps approximately We say that x causes M to carry out the k cel l
U
transformation U with accuracy in time T if for every j i V oninputj ijx ij i M halts in exactly T
U
steps with its tapehead back in the start cel l and with nal superposition U j i jxi where U is a unitary
transformation on V such that kU U k Moreover for a family A of transformations we say that M
carries out these transformations in polynomial time if T is boundedbya polynomial in and the length
of the input
In the case that A contains transformations that are not unitary we say that M carries out the set of
transformations A with closeness factor c if for any and any U A which is c close to unitary
there is a unitary transformation U with kU U k such that jx ij i causes M to carry out the
U
transformation U in time which is polynomial in and the length of its input
Recall that a near trivial transformation written as x y calls for a rotation b etween dimensions x
i
and y of angle if x y and a phase shift of e to dimension x otherwise So wewant to build a
stationary normal form QTM that takes as input w x y and transforms w according to the near
trivial transformation describ ed by x y with accuracy We also need this QTM s running time to
dep end only on the length of w but not its value If this is the case then the machine will also halt on an
initial sup erp osition of the form j ijx y ij i where j i is a sup erp osition of equal length strings w
with rst track alphabet f g that carries Lemma There is a stationary normal form QTM M
out the set of near trivial transformations on its rst trackinpolynomial time
Pro of Using the enco ding x y for near trivial transformations describ ed ab ovein x we will show
how to construct QTMs M and M with rst track alphab et f g such that M carries out the set of
near trivial rotations on its rst track in p olynomial time and M carries out the set of near trivial phase
shifts on its rst track in p olynomial time Using the Branching Lemma page pagerefbranching lemma
on these twomachines we can construct a QTM that b ehaves correctly provided there is an extra track
containing a when x y and wehave a phase shift to p erform and containing a when x y and we
have a rotation to p erform We can then construct the desired QTM M bydovetailing b efore and after
with machines that compute and erase this extra bit based on x y The Synchronization Theorem lets us
construct two such stationary normal form QTMs whose running times dep end only on the lengths of x
and y Therefore this additional computation will not disturb the synchronization of the computation
Next we showhow to construct the QTM M to carry out near trivial rotations
It is easy to construct a QTM which on input b applies a rotation by angle between j i and j i while
leaving b alone if b But Lemma ab ove tells us that wecanachieveany rotation by applying
the single rotation R at most a p olynomial numb er of times Therefore the following ve step pro cess will
allow us to apply a near trivial rotation Wemust b e careful that when we apply the rotation the two
computational paths with b f g di er only in b since otherwise they will not interfere
Calculate k such that k Rmo d
Transform w x y into b x y z where b if w x b ifw y and w x and b otherwise
and where z w if b and z is the empty string otherwise
Run the rotation applying machine k times on the rst bit of z
Reverse Step transforming x y w with w x y into w x y transforming x y into x x y and
transforming x y with x y into y x y
Reverse Step erasing k
h of these ve steps and then dovetailing We build the desired QTM M by constructing a QTM for eac
them together
First notice that the length of the desired output of Steps and can b e computed just from the
length of the input Therefore using Lemmas from page and the Synchronization Theorem from
page we can build p olynomial time stationary normal form QTMs for Steps and which run in
time which dep ends on the lengths of w x y but not their particular values
To complete the construction wemust build a machine for the rotation with these same prop erties
The stationary normal form QTM R with alphab et f g state set fq q q g and transition function
f
de ned by
q j ijq ijLi cos Rj ijq ijLi sin Rj ijq ijLi
sinRj ijq ijLi cosRj ijq ijLi
q j ijq ijRi
f
q j ijq ijRi j ijq ijRi j ijq ijRi
f
runs for constant time and applies rotation R between start cell contents j i and j i while leaving other
inputs unchanged Inserting R for the sp ecial state in the reversible TM from the Lo oping Lemma we
can construct a normal form QTM which applies rotation k R between inputs j ki and j ki while leaving
input j w ki unchanged Since the machine we lo op on is stationary and takes constant time regardless
of its input the resulting lo oping machine is stationary and takes time dep ending only on k
Finally with appropriate use of Lemmas on page and on page we can dovetail these
ve stationary normal form QTMs to achieve a stationary normal form QTM M that implements the
desired computation Since the phase application machine run in time which is indep endentofw x and
y and the other four run in time which dep ends only on the length of w x and y the running time of M
dep ends on the length of w x andy but not on their particular values Therefore it halts with the prop er
output not only if run on an input with a single string w but also if run on an initial sup erp osition with
di erent strings w but with the same near trivial transformation and
The QTM M to carry out near trivial phase shifts is the same as M except that we replace the
transition function of the simple QTM which applies a phase shift rotation by angle R as follows giving a
iR
stationary QTM which applies phase shift e if b is a and phase shift otherwise
iR
q j ijq ijLi e j ijq ijLi j ijq ijLi
q j ijq ijRi
f
j ijq ijRi j ijq ijRi j ijq ijRi q
f
The pro of that this gives the desired M is identical to the pro of for M and is omitted
Carrying out a unitary transformation on a QTM
Now that we can approximate a unitary transformation by a pro duct of near trivial transformations and
wehave a QTM to carry out the latter we can build a QTM to apply an approximation of a given unitary
transformation
Theorem Unitary Transformation Theorem There is a stationary normal form QTM M
with rst track alphabet f g that carries out the set of al l transformations on its rst track in poly
p
nomial time with required closeness factor for transformations of dimension d
d
d
k
p
Pro of Given an and a transformation U of dimension d whichis close to unitary we
d
d
can carry out U to within on the rst k cells of the rst track using the following steps
and a list of near trivial U U such that kU U U k Calculate and write on clean tracks
n n
n
k
and such that n is p olynomial in
Apply the list of transformations U U each to within
n
n
Erase U U and
n
n
We can construct a QTM to accomplish these steps as follows First using Lemma on page
and the Synchronization Theorem on page we can build p olynomial time stationary normal form
QTMs for Steps and which run in time which dep ends only on U and Finally we can build a
k
stationary normal form QTM to accomplish Step in time which is p olynomial in and as follows
Wehave a stationary normal form QTM constructed in Lemma on page to apply any sp eci ed
near trivial transformation to within a given b ound Wedovetail this with a machine constructed using
the Synchronization Theorem on page that rotates U around to the end of the list of transformations
Since the resulting QTM is stationary and takes time which dep ends only on and the U we can insert
i
it for the sp ecial state in the machine of the Lo oping Lemma to give the desired QTM for Step
With appropriate use of Lemmas on page and on page dovetailing the QTMs for
these three steps gives the desired M Since the running times of the three QTMs are indep endentofthe
contents of the rst track so is the running time of M Finally notice that when we run M we apply to
the rst k cells of the rst track we apply a unitary transformation U with
kU U k kU U U k kU U U k n
n n
n
as desired
Constructing a universal QTM
A universal QTM must inevitably decomp ose one step of the simulated machine using many simple steps
Astepofthesimulated QTM is a mapping from the computational basis to a new orthonormal basis A
single step of the universal QTM can only map some of the computational basis vectors to their desired
destinations In general this partial transformation will not b e unitary b ecause the destination vectors will
not b e orthogonal to the computational bases whichhavenotyet b een op erated on The key construction
that enables us to achieve a unitary decomp osition is the Unidirection Lemma Applying this lemma we
get a QTM whose mapping of the computational basis has the following prop erty there is a decomp osition
of the space into subspaces of constant dimension such that each subspace gets mapp ed onto another
More sp eci cally wesaw in the previous section that we can construct a QTM that carries out to a
close approximation any sp eci ed unitary transformation On the other hand wehave also noted that
the transition function of a unidirectional QTM sp eci es a unitary transformation from sup erp ositions of
current state and tap e symb ol to sup erp ositions of new symb ol and state This means a unidirectional
QTM can b e simulated by rep eatedly applying this xed dimensional unitary transformation followed by
the reversible deterministic transformation that moves the simulated tap e head according to the new state
So our universal machine will rst convert its input to a unidirectional QTM using the construction of the
Unidirection Lemma and then simulate this new QTM by rep eatedly applying this unitary transformation
and reversible deterministic transformation
Since we wish to construct a single machine that can simulate every QTM wemust build it in such
away that every QTM can b e provided as input Much of the de nition of a QTM is easily enco ded
We can write down the size of the alphab et with the rst symb ol assumed to b e the blank symb ol
and we can write down the size of the state set Q with the rst state assumed to b e the start state To
complete the sp eci cation we need to describ e the transition function by giving the card card Q
amplitudes of the form i i i i d If we had restricted the de nition of a QTM to include only
machines with rational transition amplitudes then we could write down each amplitude explicitly as the
ratio of twointegers However wehave instead restricted the de nition of a QTM to include only those
machines with amplitudes in C which means that for each amplitude there is a deterministic algorithm
n
which computes the amplitude s real and imaginary parts to within in time p olynomial in n We will
therefore sp ecify by giving a deterministic algorithm that computes each transition amplitude to within
n
in time p olynomial in n
Since the universal QTM we will construct returns its tap e head to the start cell after simulating each
step of the desired machine it will incur a slowdown which is at least linear in T We conjecture that
with more care a universal QTM can b e constructed whose slowdown is only p olylogarithmic in T
Theorem There is a normal form QTM M such that for any wel l formedQTMM any and
M can simulate M with accuracy for T steps with slowdown polynomial in T and any T
Pro of As describ ed ab ove our approach will b e rst to use the construction of the Unidirection Lemma to
build a unidirectional QTM M whichsimulates M with slowdown by a factor of and then to simulate M
We will rst describ e the simulation of M and then return to describ e the easily computable prepro cessing
that needs to b e carried out
So supp ose M Q is a unidirectional QTM that wewishtosimulate on our universal QTM
We start by reviewing the standard technique of representing the con guration of a target TM on the
tap e of a universal TM We will use one track of the tap e of our universal QTM to simulate the current
con guration of M Since the alphab et and state set of M could haveany xed size we will use a series of
log card Q cells of our tap e referred to as a sup ercell to simulate each cell of M Each sup ercell
holds a pair of integers p where card represents the contents of the corresp onding cell of M
and p card Q represents the state of M if its tap e head is scanning the corresp onding cell and p
otherwise Since the tap e head of M can only move distance T away from the start cell in time T we only
need sup ercells for the T cells at the center of M s tap e and we place markers to denote the ends
Nowweknowthatifwe ignore the up date direction then gives a unitary transformation U of
dimension d card Q from sup erp ositions of current state and tap e symb ol to sup erp ositions of
new state and symbol So we can prop erly up date the sup erp osition on the simulation tracks if we rst
apply U to the current state and symbol of M and then move the new state sp eci cation left or rightone
sup ercell according to the direction in which that state of M can b e entered
We will therefore build a QTM ST EP that carries out one step of the simulation as follows In addition
to the simulation track this machine will is provided as input a desired accuracy a sp eci cation of U
card Q
p
which is guaranteed to b e close to the unitary and a string s f g whichgives the
d
d
direction in which eachstateofM can b e entered The machine ST EP op erates as follows
Transfer the current state and symbol p to emptywork space near the start cell leaving a sp ecial
marker in their place
Apply U to p to within transforming p into a sup erp osition of new state and symbol q
Reverse Step transferring q back to the marked empty sup ercell and emptying the work space
th
Transfer the state sp eci cation q one sup ercell to the right or left dep ending whether the q bit of
s isa or
Using the Synchronization Theorem on page we can construct stationary normal form QTMs for
Steps and whichtake time whichispolynomialinT and for a xed M dep ends only on T Step
can b e carried out in time p olynomial in card card Q and with the unitary transformation applying
QTM constructed in the Unitary Transformation Theorem With appropriate use of Lemmas on
page and on page dovetailing these four normal form QTMs gives us the desired normal form
QTM ST EP
Since each of the four QTMs takes time which dep ends for a xed M only on T and so do es ST EP
Therefore if we insert ST EP for the sp ecial state in the reversible TM constructed in the Lo oping Lemma
and provide additional input T the resulting QTM ST EP will halt after time p olynomial in T and
after simulating T steps of M with accuracy T
Finally we construct the desired universal QTM M bydovetailing ST EP after a QTM which carries
out the necessary prepro cessing In general the universal machine must simulate QTMs which are not
unidirectional So the prepro cessing for desired QTM M desired input x and desired simulation accuracy
consists of rst carrying out the construction of Unidirection Lemma to build a unidirectional QTM M
which simulates M with slowdown by a factor of The following inputs are then computed for ST EP
The prop er T sup ercell representation of the initial con guration of M with input x
p
The d dimensional transformation U for M with eachentry written to accuracy
d
T d
The string of directions s for M
The desired number of simulation steps T and the desired accuracy T
It can b e veri ed that each of these inputs to M can b e computed in deterministic time whichis
p olynomial in T and the length of the input If the transformation U is computed to the sp eci ed
p
of the desired unitary accuracy the transformation actually provided to ST EP will b e within
d
T d
U and so will b e close to unitary as required for the op eration of ST EP So each time ST EP
p
d
T d
applies runs with accuracy T it will have applied a unitary transformation which is within T
of U Therefore after T runs of ST EP wewillhave applied a unitary transformation which is within
of the T step transformation of M This means that observing the simulation trackofM after
it has completed will give a sample from a distribution which is within total variation distance of the
distribution sampled from by observing M on input x at time T
The computational p ower of QTMs
In this section we explore the computational p ower of QTMs from a complexity theoretic p oint of view It
is natural to de ne quantum analogues of classical complexity classes In classical complexity theory
BPP is regarded as the class of all languages that are e ciently computable on a classical computer
The quantum analog of BP P BQP b ounded error quantum p olynomial time should similarly b e
regarded as the class of all languages that are e ciently computable on a QTM
Accepting languages with QTMs
De nition Let M be a stationary normal form multi track QTM M whose last track has alphabet
f g IfwerunM with string x on the rst track and the empty string elsewhere wait until M halts
and then observe the last track of the start cel l we wil l seea with some probability p We wil l say that
M accepts x with probability p and rejects x with probability p
Consider any language L
We say that QTM M exactly accepts the L if M accepts every string x L with probability and
rejects every string x L with probability
We de ne the class EQP exact or error free quantum p olynomial time as the set of languages which
are exactly acceptedbysomepolynomial time QTM Moregeneral ly we de ne the class EQTime T n
Note that the transition function of M is again sp eci ed with a deterministic algorithm which dep ends on the algorithm
for the transition function of M
This can b e accomplished by p erforming a measurementtocheck whether the machine is in the nal state q Making
f
this partial measurementdoesnothaveany other e ect on the computation of the QTM
as the set of languages which are exactly accepted by some QTM whose running time on any input of length
n is boundedby T n
A QTM accepts the language L with probability p if M accepts with probability at least
p every string x L and rejects with probability at least p every string x L We de ne
the class BQP b ounded error quantum p olynomial time as the set of languages which areacceptedwith
probability by some polynomial time QTM Moregeneral ly we de ne the class BQTime T n as the
set of languages which areaccepted with probability by some QTM whose running time on any input
of length n is boundedby T n
The limitation of considering only stationary normal form QTMs in these de nitions is easily seen not
to limit the computational p ower by app ealing to the stationary normal form universal QTM constructed
in Theorem on page
Upp er and lower b ounds on the p ower of QTMs
Clearly EQP BQP Since reversible TMs are a sp ecial case of QTMs Bennett s results imply that
P EQP and BPP BQP We include these two simple pro ofs for completeness
Theorem P EQP
Pro of Let L b e a language in P Then there is some p olynomial time deterministic algorithm that
on input x pro duces output if x L and otherwise App ealing to the Synchronization Theorem on
page pagerefdet synchronized there is therefore a stationary normal form QTM running in p olynomial
time which on input x pro duces output x if x Land x otherwise This is an EQP machine accepting
L
Theorem BPP BQP
Pro of Let L b e a language in BPP Then there must b e a p olynomial p n and a p olynomial time
deterministic TM M with output which satisfy the following For anystringx of length n ifwe call
p n p n
S the set of bits computed by M on the inputs x y with y f g then the prop ortion of s in
x
S is at least when x L and at most otherwise We can use a QTM to decide whether a string
x
x in the language L by rst breaking into a sup erp osition split equally among all jxijy i and then running
this deterministic algorithm
p n
First wedovetail a stationary normal form QTM whichtakes input x to output x with a
stationary normal form QTM constructed as in Theorem b elow which applies a Fourier transform to
the contents of its second track This gives us a stationary normal form QTM which on input x pro duces
P
jxijy i Dovetailing this with a synchronized normal form version of the sup erp osition
pn
pn
y f g
M built according to the Synchronization Theorem on page pagerefdet synchronized gives a p olynomial
time QTM which on input x pro duces a nal sup erp osition
X
jxijy ijM x y i
p n
pn
y f g
Since the prop ortion of s in S is at least ifx L and at most otherwise observing the bit on
x
the third track will give the prop er classi cation for string x with probability at least Therefore this
is a BQP machine accepting the language L
Clearly BQP is in exp onential time The next result gives the rst non trivial upp er b ound on BQP
Theorem BQP PSPACE
Pro of Let M Q bea BQP machine with running time p n
According to Theorem on page anyQTMM whichis
close
card card Q p n
to M will simulate M for p n steps with accuracy If we simulate M with accuracy then the
success probability will still b e at least Therefore we need only work with the QTM M where each
transition amplitude from M is computed to its rst log card card Q p n bits
Now the amplitude of any particular con guration at time T is the sum of the amplitudes of each
p ossible computational path of M of length T from the start con guration to the desired con guration
The amplitude of eachsuch path can b e computed exactly in p olynomial time So if we maintain a stack
of at most p n intermediate con gurations we can carry out a depth rst search of the computational tree
to calculate the amplitude of any particular con guration using only p olynomial space but exp onential
time
Finally we can determine whether a string x of length n is accepted by M by computing the sum of
squared magnitudes at time p n of all con gurations that M can reach that havea inthestart cell and
comparing this sum to Clearly the only reachable accepting con gurations of M are those with a
in the start cell and blanks in all but the p n cells within distance p n of the start cell So using only
p olynomial space we can step through all of these con gurations computing a running sum of the their
squared magnitudes
P
Following Valiant s suggestion the upp er b ound can b e further improved to P This pro of
can b e simpli ed by using a theorem from whichshows howany BQP machine can b e turned into a
clean version M that on input x pro duces a nal sup erp osition with almost all of its weightonx M x
where M x isa ifM accepts x and a otherwise This means we need only estimate the amplitude of
this one con guration in the nal sup erp osition of M
Now the amplitude of a single con guration can b e broken down not only into the sum of the amplitudes
of all of the computational paths that reach it but also into the sum of p ositive real contributions the
sum of negative real contributions the sum of p ositive imaginary contributions and the sum of negative
imaginary contributions We will show that each of these four pieces can b e computed using a P machine
Recall that P is the set of functions f mapping strings to integers for which there exists a p olynomial
p n and a language L P such that for any string x the value f x is the numb er of strings y of length
p jxj for which xy is contained in L
Theorem If the language L is contained in the class BQTime T n with T n n with T n
time constructible then for any thereisaQTMM which accepts L with probability and has
the fol lowing property When run on input x of length n M runs for time boundedbycT n where c is
es a nal superposition in which jxijL x i with L x if x L and apolynomial in log andproduc
otherwise has squared magnitude at least
P
Theorem BQP P
Pro of Let M Q bea BQP machine with observation time p n App ealing to Theorem
ab ove we can without loss of generality that M is a clean BQP machine i e on any input x at time
p jxj the squared magnitude of the nal con guration with output x M x isatleast
Nowasabove we will app eal to Theorem on page which tells us that if wework with the
QTM M where each amplitude of M is computed to its rst b log card card Q p n bits and
we run M on input x then the squared magnitude at time p jxj of the nal con guration with output
x M x will b e at least
We will carry out the pro of by showing how to use an oracle for the class P to e ciently compute
the amplitude of the nal con guration x ofM at time T Since with error magnitude less than
the true amplitude has magnitude at most the squared magnitude of this approximated amplitude must
b e within of the squared magnitude of the true amplitude To see this just note that if is the true
amplitude and k k then
k k k k k k k kk k
Since the success probabilityofM is at least comparing the squared magnitude of this approximated
amplitude to lets us correctly classify the string x
We will now showhow to approximate the amplitude describ ed ab ove First notice that the amplitude
of a con guration at time T is the sum of the amplitudes of each computational path of length T from the
start con guration to the desired con guration The amplitude of any particular path is the pro duct of the
amplitudes in the transition function of M used in each step along the path Since each amplitude consists
T
of a real part and an imaginary part we can think of this pro duct as consisting of the sum of terms
each of which is either purely real or purely imaginary So the amplitude of the desired con guration at
T
time T is the sum of these terms over each path We will break this sum into four pieces the sum of
the p ositive real terms the sum of the negative real terms the sum of the p ositive imaginary terms and
the sum of the negative imaginary terms and we will compute each to within error magnitude less than
with the aid of a P algorithm Taking the di erence of the rst two and the last two will then give
us the amplitude of the desired con guration to with error of magnitude at most as desired
We can compute the sum of the p ositiverealcontributions for all paths as follows Supp ose for some
xed constant c which is p olynomial in T we are given the following three inputs all of length p olynomial
T
in T a sp eci cation of a T step computational path p of M a sp eci cation t of one of the terms
cT
and an integer w between and Then it is easy to see that we could decide in deterministic time
p olynomial in T whether p is really a path from the start con guration of M on x to the desired nal
th
con guration whether the term t is real and p ositive and whether the term t term of the amplitude for
cT
path p is greater than w Ifwe xapathp and term t satisfying these constraints then the number
cT cT th
of w for which this algorithm accepts divided by is within of the value of the t for path p
So if we x only a path p satisfying these constraints then the number of t w for which the algorithm
cT c T
accepts divided by is within of the sum of the p ositive real terms for path p Therefore
cT c T
the number of p t w for which the algorithm accepts divided by is within N of the sum of
all of the p ositive real terms of all of the T step paths of M from the start con guration to the desired
con guration Since there are at most card card Q p ossible successors for any con guration in a legal
card card Q logT
log
c T
path of M cho osing c gives N as desired Similar
T T
P
reasoning gives P algorithms for approximating each of the remaining three sums of terms
Oracle QTMs
In this subsection and the next we will assume without loss of generality that the TM alphab et for each
trackis f g Initially all tracks are blank except that the input trackcontains the actual input
surrounded by blanks We will use to denote f g
In the classical setting an oracle may b e describ ed informally as a device for evaluating some Bo olean
function f on arbitrary arguments at unit cost p er evaluation This allows to formulate
questions suchas iff were e ciently computable by a TM which other functions or languages could
b e e ciently computed by TMs In this section we de ne oracle QTMs so that the equivalent question
can b e asked in the quantum setting
An oracle QTM has a sp ecial query track on which the machine will place its questions for the oracle
Oracle QTMs havetwo distinguished internal states a pre query state q and a p ost query state q
q a
A query is executed whenever the machine enters the pre query state with a single non empty blo ckof
non blank cells on the query track Assume that the non blank region on the query tap e is in state jx bi
when the pre query state is entered where x b and denotes concatenation Let f be the
Bo olean function computed by the oracle The result of the oracle call is that the state of the query tap e
b ecomes jx b f x i where denotes the exclusive or addition mo dulo after which the machine s
internal control passes to the p ost query state Except for the query tap e and internal control other parts
of the oracle QTM do not change during the query If the target bit jbi was supplied in initial state j i
then its nal state will b e jf x i just as in a classical oracle machine Conversely if the target bit is
already in state jf x i calling the oracle will reset it to j i a pro cess known as uncomputing whichis
essential for prop er interference to take place
The p ower of quantum computers comes from their ability to follow a coherent sup erp osition of com
putation paths Similarly oracle QTMs derive great p ower from the ability to p erform sup erp ositions of
queries For example an oracle for Bo olean function f might b e called when the query tap e is in state
P
j i jx i where are complex co e cients corresp onding to an arbitrary sup erp osition of
x x
x
queries with a constant j i in the target bit In this case after the query the query string will b e left in
P
the entangled state jx f x i
x
x
That the ab ove de nition of oracle QTMs yields unitary evolutions is self evidentifwe restrict ourselves
to machines that are well formed in other resp ects in particular evolving unitarily as they enter the pre
query state and leave the p ost query state
O
Let us de ne BQTime T n as the sets of languages accepted with probability at least bysome
O
oracle QTM M whose running time is b ounded by T n This b ound on the running time applies to
O
each individual input not just on the average Notice that whether or not M is a BQP machine might
O O
might not b e one dep end up on the oracle O thus M mightbeaBQP machine while M
Wehave carefully de ned oracle QTMs so that the same technique used to reverse a QTM in the
Reversal Lemma can also b e used to reverse an oracle QTM
If M is a normal form unidirectional oracle QTM then there is a normal form oracle Lemma
O O
QTM M such that for any oracle O M reverses the computation of M while taking two extratime
steps
Pro of Let M Q b e a normal form unidirectional oracle QTM with initial and nal states q
and q and with query states q q and q We construct M from M exactly as in the pro of of the
f q f a
Since a QTM must b e careful to avoid leaving around intermediate computations on its tap e requiring that the query
track contain only the query string adds no further di culty to the construction of oracle QTMs
Reversal Lemma We further give M the same query states q q as M but with roles reversed Since M
q a
is in normal form and q q wemust have q q Recall that the transition function of M is de ned
q f a
so that for q q
X
q p q d j ijpijd i
q p
p
Therefore since state q always leads to state q in M stateq always leads to state q in M Therefore
q a a q
M is an oracle QTM
Next note that since the tap e head p osition is irrelevant for the functioning of the oracle the op eration
O O
of the oracle in M reverses the op eration of the oracle in M Finally the same argument used in the
O O
Reversal Lemma can b e used again here to prove that M is well formed and that M reverses the
O
computation of M while taking two extra time steps
The ab ove de nition of a quantum oracle for an arbitrary Bo olean function will su ce for the purp oses
of the present pap er but the abilityofquantum computers to p erform general unitary transformations
suggests a broader de nition whichmaybeusefulinothercontexts For example oracles that p erform
more general non Bo olean unitary op erations have b een considered in computational learning theory
and used to obtain large separations b etween quantum and classical relativized complexity classes
See for a discussion of more general de nitions of oracle quantum computing
Fourier Sampling and the p ower of QTMs
In this section wegive evidence that QTMs are more p owerful than b ounded error probabilistic TMs We
de ne the recursiveFourier sampling problem which on input the program for a b o olean function takes on
value or Weshow that the recursiveFourier sampling problem is in BQP On the other hand we
prove that if the b o olean function is sp eci ed by an oracle then the recursiveFourier sampling problem
o log n
is not in BQTime n This result provided the rst evidence that QTMs are more p owerful than
classical probabilistic TMs with b ounded error probability
One could ask what the relevance of these oracle results is in view of the non relativizing results on
probabilistically chec kable pro ofs Moreover Arora et al make the case that the fact that the P
versus NP question relativizes do es not imply that the question cannot b e resolved by currenttechniques
in complexity theory On the other hand in our oracle results and also in the subsequent results of
the key prop erty of oracles that is exploited is their back b ox nature and for the reasons sketched b elow
these results did indeed provide strong evidence that BQP BPP of course the later results of
gaveeven stronger evidence This is b ecause if one assumes that P NP and the existence of one way
functions b oth these hyp otheses are unproven but widely b elieved by complexity theorists it is also
reasonable to assume that in general it is imp ossible to e ciently gure out the function computed by
a program by just lo oking at its text i e without explicitly running it on various inputs Such a program
would have to b e treated as a black b ox Of course such assumptions cannot b e made if the question at
hand is whether P NP
n
C There is a Fourier Sampling Consider the vector space of complex valued functions f Z
natural inner pro duct on this space given by
X
f x g f g x
n
x Z
n
The standard orthonormal basis for the vector space is the set of delta functions Z C given by
y
y and x for x y Expressing a function in this basis is equivalent to sp ecifying its values
y y
n
at each p oint in the domain The characters of the group Z yield a di erent orthonormal basis consisting
P
s x
n
n
of the parity basis functions Z C given by x where s x s x Given any
s s i i
n
i
P
n n
function f Z C wemay write it in the new parity basis as f f s where f Z C is
s
s
given by f s f The function f is called the discrete Fourier transform or Hadamard transform
s
of f Here the latter name refers to the fact that the linear transformation that maps a function f to its
Fourier transform f is the Hadamard matrix H Clearly this transformation is unitary since it is just
n
e ecting a change of basis from one orthonormal basis the delta function basis to another the parity
P P
s j function basis It follows that jf x j jf
x s
n
Moreover the discrete Fourier transform on Z can b e written as the Kronecker pro duct of the trans
p p
form on eachofn copies of Z the transform in that case is given by the matrix This
p p
fact can b e used to write a simple and e cient QTM that e ects the discrete Fourier transformation in the
following sense supp ose the QTM has tap e alphab et f g and the initially only n tap e cells contain
P
non blank symb ols Then we can express the initial sup erp osition as f x jxi where f x isthe
n
x f g
amplitude of the con guration with x in the non blank cells Then there is a QTM that e ects the Fourier
transformation by applying the ab ove transform on eachofthe n bits in the n non blank cells This takes
P
f s jsi Theorem b elow Notice that this O n steps and results in the nal con guration
n
s f g
n
fourier transformation is taking place over a vector space of dimension On the other hand the result
of the fourier transform f resides in the amplitudes of the quantum sup erp osition and is thus not
directly accessible Wecan however p erform a measurement on the n non blank tap e cells to obtain
P such that P s jf s j We shall refer to this op eration a sample from the probability distribution
as Fourier sampling As we shall see this is a very p owerful op eration The reason is that eachvalue
f s dep ends up on all the exp onentially manyvalues of f and it do es not seem p ossible to anticipate
whichvalues of s will have large probability constructiveinterference without actually carrying out an
exp onential search
n n
Given a b o olean function g Z f g wemay de ne a function f Z C of norm by letting
n
f x g x Following Deutsch and Josza if g is a p olynomial time computable function there
P
is a QTM that pro duces the sup erp osition g x jxi in time p olynomial in n Combining this
n
x f g
with the Fourier sampling op eration ab ove we get a p olynomial time QTM that samples from a certain
distribution related to the given p olynomial time computable function g Theorem b elow We shall
call this comp osite op eration Fourier sampling with resp ect to g
Theorem There is a normal form QTM which when run on an initial superposition of n bit strings
j i halts in time n with its tapehead back in the start cel l and produces nal superposition H j i
n
Pro of We can construct the desired machine using the alphab et f g and the set of states fq q q q q g
a b c f
The machine will op erate as follows when started with a string of length n as input In state q the ma
hine steps right along the input string chine steps right and left and enters state q In state q the mac
b b
until it reaches the at the end and enters state q stepping back left to the string s last symb ol During
c
p p
to the bit in each cell In state this rightward scan the machine applies the transformation
p p
q the machine steps left until it reaches the to the left of the start cell at which p oint steps backright
c
and halts The following gives the quantum transition function of the desired machine
q j ijq ijRi j ijq ijRi
a a
q j ijq ijLi
a b
p p p p
j ijq ijRi j ijq ijRi j ijq ijRi j ijq ijRi j ijq ijLi q
b b b b c b
q j ijq ijLi j ijq ijLi j ijq ijLi
c f c c
j ijq ijRi j ijq ijRi j ijq ijRi q
f
It is can b e easily veri ed that the Completion Lemma can b e used to extend this machine to a well formed
QTM and that this machine has the desired b ehavior describ ed ab ove
n
Theorem For every polynomial time computable function g f g f g thereisapolyno
n
mial time stationary QTM where observing the nal superposition on input gives each n bit string s
n
with probability jf s j where f x g x
Pro of We build a QTM to carry out the following steps
P
n
Apply a F ourier transform to to pro duce jii
n
n
i f g
P
Compute f to pro duce jiijf i i
n
n
i f g
P
f i jiijf i i Apply a phase applying machine to pro duce
n
n
i f g
P
Apply the reverse of the machine in Step to pro duce f i jii
n
n
i f g
P
f s jsi as desired Apply another Fourier transform to pro duce
n
n
i f g
Wehave already constructed ab ove the Fourier transform machine for Steps and Since f can b e
computed in deterministic p olynomial time the Synchronization Theorem on page lets us construct
p olynomial time QTMs for Steps and whichtake input jii to output jiijf i i and vice versa with
running time indep endent of the particular value of i
Finally we can apply phase according to f i in Step by extending to two tracks the stationary
normal form QTM with alphab et f g de ned by
q j ijq ijRi j ijq ijRi j ijq ijRi
q j ijq ijLi j ijq ijLi j ijq ijLi
f f f
j ijq ijRi j ijq ijRi j ijq ijRi q
f
Dovetailing these ve stationary normal form QTMs gives the desired machine
Remarkably this quantum algorithm p erforms Fourier sampling with resp ect to f while calling the
algorithm for f only twice To see more clearly why this is remarkable consider the sp ecial case of
the sampling problem where the function f is one of the unnormalized parity functions Supp ose f
i k n
corresp onds to the parity function i e f i where i k f g Then the result of fourier
k
sampling with resp ect to f is always k Let us call this promise problem the parity problem on input
a program that computes f where f is an unnormalized parity function determine k Notice that the
QTM for fourier sampling extracts n bits of information the value of k using just twoinvo cations of the
subroutine for computing b o olean function f It is not hard to showthatifwe f is sp eci ed by an oracle
then in a probabilistic setting extracting n bits of information must require at least n invo cations of the
b o olean function Wenowshowhow to amplify this this advantage of quantum computers over probabilistic
computers by de ning a recursiveversion of the parity problem the recursiveFourier sampling problem
To ready this problem for recursion we rst need to turn the parity problem into a problem with range
f g This is easily done by adding a second function g and requiring the answer g k rather than k
itself
n
Now we will make the problem recursive For each problem instance of size n we will replace the
n
values of f with indep endent recursive subproblems of size n and so on stopping the recursion with
function calls at the b ottom Since a QTM needs only two calls to f to solve the parity problem it will
b e able to solve an instance of the recursive problem of size n recursively in time T n where
T n pol y n T n T n pol y n
However since a PTM needs n calls to f to solve the parity problem the straightforward recursive solution
to the recursive problem on a PTM will require time T n where
log n
T n nT n T n n
To allow ustoprove a separation b etween quantum and classical machines we will replace the functions
with an oracle For any legal oracle O any oracle satisfying some sp ecial constraints to b e describ ed
below we will de ne the language R We will show that there is a QTM whichgiven access to any legal
O
oracle O accepts R in time O n log n with success probability but that there is a legal oracle O such
O
o log n O
that R is not contained in BPTime n
O
Our language will consist of strings from f g with length a p ower of We will call all such strings
candidates The decision whether a candidate x is contained in the language will dep end on a recursive
n
tree If candidate x has length n then it will have children in the tree and in general a no de at
l
level l counting from the b ottom with leaves at level for convenience will have children So
n
the ro ot of the tree for a candidate x of length n is identi ed by the string x its children are each
n
identi ed by a string x x where x f g and in general a descendent in the tree at level l is
l
identi ed by a string of the form x x x x with jxj n jx j n jx j Notice that
m m
any string of this form for some n apower of and l
in log n de nes a no de in some tree So we will consider oracles O which map queries over the
alphab et f g to answers f g and we will use eachsuch oracle O to de ne a function V that
O
gives eachnodeavalue f g The language R will contain exactly those candidates x for which
O
V x
O
We will de ne V for leaves level no des based directly on the oracle O So ifx is a leaf then we
O
let V x O x
O
We will de ne V for all other no des by lo oking at the answer of O to a particular query whichis
O
chosen dep ending on the values of V for its children Consider no de x at level l We will insist that
O
l
the oracle O b e such that there is some k f g such that the children of x all havevalues determined
x
by their paritywithk
x
l
y k
x
V x y y f g
O
and we will then give V x the value O x k We will say that the oracle O is legal if this pro cess allows
O x
us to successfully de ne V for all no des whose level is Any query of the form x k with x anodeat
O
l
level l and k f g or of the form x with x a leaf is called a query at node x A query whichis
lo cated in the same recursive tree as no de x but not in the subtree ro oted at x is called outside of x
Notice that for any candidate x the values of V at no des in the tree ro oted at x and the decision whether
O
x is in R all dep end only on the answers of queries lo cated at no des in the tree ro oted at x
O
O
Theorem Thereisanoracle QTM M such that for every legal oracle O M runs in polynomial
time and accepts the language R with probability
O
Pro of Wehave built the language R so that it can b e accepted e ciently using a recursive quantum
O
algorithm Toavoid working through the details required to implement a recursive algorithm reversibly
we will instead implement the algorithm with a machine that writes down an iteration that unwraps the
desired recursion Then the Lo oping Lemma will let us build a QTM to carry out this iteration
First consider the recursive algorithm to compute V for a no de x Ifx is a leaf then the value V x
O O
can b e found by querying the string x If x is a no de at level l then calculate V as follows
O
l
Split into an equal sup erp osition of the children of x
Recursively compute V for these children in sup erp osition
O
Apply phase to eachchild given bythatchild s value V
O
Reverse the computation of Step to erase the value V
O
Apply the Fourier transform converting the sup erp osition of children of x into a sup erp osition con
sisting entirely of the single string k
x
Query x k to nd the value V for x
x O
Reverse Steps to erase k leaving only x and V x
x O
Notice that the numb er of steps required in the iteration ob eys the recursion discussed ab ove and hence
is p olynomial in the length of x So we can use the Synchronization Theorem on page to construct a
p olynomial time QTM which for any particular x writes a list of the steps whichmust b e carried out
Therefore we complete the pro of by constructing a p olynomial time QTM to carry out any such a list of
step
Since our algorithm requires us to recursively run b oth the algorithm and its reverse we need to see
how to handle each step and its reverse Before westart we will ll out the no de x to a description
of a leaf by adding strings of s Then Steps and at level l just require applying the Fourier
l
transform QTM to the bit string at level l in the current no de description Since the Fourier transform
is its own reverse the same machine also reverses Steps and Step is handled by the phase applying
machine already constructed ab oveaspartoftheFourier sampling QTM in Theorem Again the
transformation in Step is its own reverse so the same machine can b e used to reverse Step Step
and its reverse can b e handled byareversible TM which copies the relevant part of the currentnode
description queries the oracle and then returns the no de description from the query tap e Notice that
each of these machines takes time which dep ends only on the length of its input
Since wehave stationary normal form QTMs to handle eachstepatlevel l and its reverse in time
l
b ounded by a p olynomial in we can use the Branching Lemma to construct a stationary normal form
QTM to carry out any sp eci ed step of the computation Dovetailing with a machine which rotates the
rst step in the list to the end and inserting the resulting machine into the reversible TM of the Lo oping
Lemma gives the desired QTM
log n
Computing the function V takes time n even for a probabilistic computer allowed a b ounded
O
probability of error We can see this with the following intuition First consider asking some set of
queries of a legal O that determine the value of V for a no de x describ ed by the string x x at
O m
level l There are twoways that the asked queries might x the value at x The rst is that the queries
outside of the subtree ro oted at x might b e enough to x V for x If this happ ens wesay that V x
O O
is xedbyconstraint An example of this is that if weasked all of the queries in the subtrees ro oted at
all of the siblings of x then wehave xed V for all of the siblings thereby xing the string k such that
O
y k
V x x y equals
O m
The only other way that the value of the no de x mightbe xedisthefollowing If the queries x the
value of V for some of the children of x then this will restrict the p ossible values for the string k such
O x
y k
x
that V x y always equals and such that V x O x k If the query x k for each p ossible
O O x x
k has b een asked and all have the same answers then this xes the value V at x If the query x k for
x O x
the correct k has b een asked then wecall x a hit
x
Now notice that xing the values of a set of children of a level l no de x restricts the value of k to a set
x
l
c
of p ossibiliti es where c is the maximum size of a linearly indep endent subset of the children whose
log n
values are xed This can b e used to prove that it takes n n n
However this intuition is not yet enough since we are interested in arguing against a probabilistic TM
log n
rather than a deterministic TM So we will argue not just that it takes n queries to x the value
log n
of a candidate of length n but that if fewer than n queries are xed then cho osing a random legal
oracle consistent with those queries gives to a candidate of length n the value with probability extremely
close to This will give the desired result To see this call the set of queries actually asked and the
answers given to those queries a run of the probabilistic TM We will haveshown that if wetakeany run
log n
on a candidate of length n with fewer than n queries then the probability that a random legal oracle
agreeing with the run assigns to x the v alue is extremely close to This means that a probabilistic
o log n
TM whose running time is n will fail with probability to accept R for a random legal oracle O
O
De nition A run of size k is de nedasapair S f where S is a set of k query strings and f is map
from S to f g such that thereisatleast one legal oracle agreeing with f
Let r be a run let y bea node at level l and let O bea legal oracle at and below y which agrees
with r Then O determines the string k for which V y O y k Ify k is a query in r then we say
y O y y
O makes y a hit for r Suppressing the dependency on r in the notation we de ne P y as the probability
that y is a hit for r when we choose a legal oracle at and below y uniformly at random from the set of al l
such oracle at and below y which agree with r Similarly for x an ancestor of y we de ne P y as the
x
probability that y is a hit when a legal oracle is chosen at and below x uniformly at random from the set of
al l oracle at and below x which agree with r
Lemma P y P y
x
Pro of Let S b e the set of legal oracles at and b elow y which agree with r We can write S as the disjoint
union of S and S where the former is the set of those oracles in S that make y a hit for r Further
h n
splitting according to the value V y we can write S as the disjoint union of four sets S S S S
O h h n n
card S
h
It is easy to see that since the oracles in S do Using this notation wehave P y
n
card S card S
n
h
not make y a hit for r card S card S card S
n n n
Next consider the set T of all legal oracles de ned at and b elow x but outside y which agree with
r Each oracle O T determines by constraint the value V y but leaves the string k completely
O y
undetermined If we again write T as the disjoint union of T and T according to the constrained value
V y we notice that the set of legal oracles at and b elow x is exactly T S T S So wehave
O
card T card S card T card S
h h
P y
x
card T card S card T card S
card T card S card T card S
h h
card T card S card T card S card T card S
h h n
Without loss of generality let card T card T Then since n n c with c n increases with n
wehave
card T card S
h
P y
x
card T card S card T card S
h n
card T card S
h
card T card S card T card S
h n
card S
h
P y
card S card S
h n
log n
For a p ositiveinteger n we de ne n n n Notice that n n
Theorem Suppose r is a run y is a node at level l with q queries from r at or below y and x
l
is an ancestor of y Then P y q n where n
x
Pro of We prove the theorem by induction on l
So x a run r and a no de y at level with q queries from r at or b elo w y Ifq y can never b e a
hit So certainly the probability that y is a hit is at most q as desired
Next we p erform the inductive step So assume the theorem holds true for any r and y at level less
than l with l Then x a run r and a no de y at level l with q queries from r at or b elow y Let
q
l
n Wewillshow that P y and then the theorem will follow from Lemma So for the
n
remainder of the pro of all probabilities are taken over the choice of a legal oracle at and b elow y uniformly
at random from the set of all those legal oracles at and b elow y which agree with r
Now supp ose that q of the q queries are actually at y Clearly ifwe condition on there b eing no
hits among the children of y then k will b e chosen uniformly among all n bit strings and hence the
y
n n
probability y is a hit would b e q Ifwe instead condition on there b eing exactly c hits among the
n c
children of y then the probability that y is a hit mustbeatmostq Therefore the probability y is
n
a hit is b ounded ab oveby the sum of q and the probability that at least n ofthechildren of y are
hits
y Applying the inductivehyp othesis with y and z taking the roles of x Now consider anychild z of
and y we know that if r has q queries at and b elow z then P z q n Therefore the expected
z y z
numb er of hits among the children of y is at most q q n This means that the probability that
at least n of the children of y are hits is at most
q q q q
n n n
Therefore
q q q q
P y
n
n n
n
since n for n
o log n
Corollary For any T n which is n relative to a random legal oracle O withprobability
R is not containedin BPTime T n
O
o log n
Pro of Fix T n whichisn
We will show that for any probabilistic TM M whenwe pick a random legal oracle O with probability
O o log n
M either fails to run in time cn or it fails to accept R with error probability b ounded by
O
Then since there are a countable numb er of probabilistic TMs and the intersection of a countable number
of probability events still has probability we conclude that with probability R is not contained in
O
o log n
BPTime n
O
We prove the corollary byshowing that for large enough n the probability that M runs in time
n
greater than T n or has error greater than on input is at least for every way of xing the
n
oracle answers for trees other than the tree ro oted at The probabilityistaken over the random choices
n
of the oracle for the tree ro oted at
n
Fix arbitrarily a legal b ehavior for O on all trees other than the one ro oted at Then consider
n O n
picking a legal b ehavior for O for the tree ro oted at uniformly at random and run M on input We
O n
can classify runs of M on input based on the run r that lists the queries the machines asks and the
answers it receives If wetake all probabilities over b oth the randomness in M and the choice of oracle O
O n
then the probabilit y M correctly classi es in time T n is
X
Pr r Pr cor r ect j r
r
where Pr r is the probabilityofrunr wherePr correct j r is the probabilitytheanswer is correct given
run r and where r ranges over all runs with at most T n queries Theorem tells us that if we
n
n queries then the probability is a hit is less than condition on any run r with fewer than
n
This means that the probability the algorithm correctly classi es conditioned on any particular run r
with fewer than n queries is at most Therefore for n large enough that T n is less than
O n
n the probability M correctly classi es in time T n is at most So for su ciently large n
O
either fails to run in time T n or has success when wecho ose O then with probability at least M
n
probability less than on input
A A QTM is well formed i its time evolution is unitary
First we note that the time evolution op erator of a QTM always exists Note that this is true even if the
QTM is not well formed
Lemma A If M is a QTM with time evolution operator U thenU has an adjoint operator U in the
inner product space of superpositions of the machine M
Pro of Let M b e a QTM with time evolution op erator U Then the adjointofU is the op erator U is
the op erator whose matrix elementinany pair of dimensions i j is the complex conjugate of the matrix
elementofU in dimensions j i The op erator U de ned in this fashion still maps all sup erp ositions to
other sup erp ositions nite linear combinations of con gurations since any particular con guration of
M can b e reached with non zero weight from only a nite numb er of other con gurations It is also easy
to see that for any sup erp ositions
hU j i h jU i
as desired
We include the pro ofs of the following standard facts for completeness We will use them in the pro of
of the theorem b elow
Fact A If U is a linear operator on an inner product space V and U exists then U preserves norm
i U U I
Pro of For any x V the square of the norm of Ux is Ux Ux x U Ux It clearly follows that if
U U I then U preserves norm For the converse let B U U I Since U preserves norm for every
x V x U Ux x x Therefore for every x V x B x It follows that B and therefore
U U I
This further implies the following useful fact
Fact A Suppose U is an linear operator in an inner product space V and U exists Then
x V kUxk kxk x y V Ux Uy x y
Pro of Since kUxk kxk Ux Ux x x one direction follows by substituting x y For the other
direction if U preserves norm then by fact A U U I Therefore Ux Uy x U Uy x y
We need to establish one one additional fact ab out norm preserving op erators b efore we can prove our
theorem
Fact A Let V bea countable inner product space and let fjiig be an orthonormal basis for V
i I
If U is a norm preserving linear operator on V and U exists then i I kU jiik Moreover if
i I kU jiik then U is unitary
Pro of Since U preserves norm kUU jiik kU jiik But the pro jection of UU jii on jii has norm
jhijUU jiij kU jiik Therefore kU jiik kU jiik and therefore kU jiik
Moreover if kU jiik then since U is norm preserving kUU jiik On the other hand the
pro jection of UU jii on jii has norm kU jiik It follows that for j i the pro jection of UU jii on
jj i must have norm Thus jhj jUU jiij Itfollows that UU I
If V is nite dimensional then U U I implies UU I and therefore an op erator is norm preserving
if and only if it is unitary However if H is in nite dimensional then U U I do es not imply UU
I Nevertheless the time evolution op erators of QTMs have a sp ecial structure and in fact these two
conditions are equivalent for the time evolution op erator of a QTM
Theorem A A QTM is wel l formed i its time evolution operator is unitary
Pro of Let U b e the norm preserving time evolution op erator of a well formed QTM M Q
Consider the standard orthonormal basis for the sup erp ositions of M given by the set of vectors jci where
c ranges over all con gurations of M as always jci is the sup erp osition with amplitude for con guration
c and elsewhere Wemay express the action of U with resp ect to this standard basis byacountable
th
dimensional matrix whose c c entry u hc jU jci This matrix has some sp ecial prop erties First each
c c
row and column of the matrix has only a nite numb er of non zero entries Secondly there are only nitely
many di erenttyp es of rows where tworows are of the same typ e if their entries are just p ermutations
of each other We shall show that eachro w of the matrix has norm and therefore by the Fact A
ab ove U is unitary Todosowewillidentify a set of n columns of the matrix for arbitrarily large n
and restrict attention to the nite matrix consisting of all the chosen rows and all columns with non zero
entries in these rows Let this matrix b e the m n matrix B By construction B satis es two prop erties
it is almost square m n for arbitrarily small There is a constant a such that each distinct
rowtyp e of the in nite matrix o ccurs at least m a times among the rows of B
Now the sum of the squared norms of the rows of B is equal to the sum of the squared norms of
the columns The latter quantity is just n since the columns of the in nite matrix are orthonormal by
Fact A ab ove If we assume that some row of the in nite matrix has norm for then we can
cho ose n su ciently large and su ciently small so that the sum of the squared norms of the rows is at
most m a m a m m a n n n a n This gives the required contradiction and
therefore all rows of the in nite matrix have norm and therefore by the Fact A ab ove U is unitary
To construct the nite matrix B let k and x some contiguous set of k cells on the tap e Consider
the set of all con gurations S such that the tap e head is lo cated within these k cells and such that the
tap e is blank outside of these k cells It is easy to see that the numb er of such con gurations n
k
k card card Q The columns indexed by con gurations in S are used to de ne the nite matrix B
referred to ab ove The non zero entries in these columns are restricted to rows indexed by con gurations
in S together with rows indexed by con gurations where the tap e head is in the cell immediately to the
left or rightofthe k sp ecial cells and such that the tap e is blank outside of these k cells The
k
numb er of these additional con gurations over and ab ove S is at most card card Q Therefore
m n k card For any wecancho ose k large enough such that m n
Recall that a row of the in nite matrix corresp onding to the op erator U is indexed by con gurations
Wesay that a con guration c is of typ e q if c is in state q Q and the three adjacent tap e cells
centered ab out the tap e head contain the three symb ols The entries of a row indexed by
c must b e a p ermutation of the entries of a row indexed byany con guration c of the same typ e as c
This is b ecause a transition of the QTM M dep ends only on the state of M and the symb ol under the tap e
head and since the tap e head moves to an adjacent cell during a transition Moreover any con guration
d that yields con guration c with non zero amplitude as a result of a single step must have tap e contents
identical to those of c in all but these three cells It follows that there are only a nite numb er of such
con gurations d and the row indexed by c can hav e only a nite numb er of non zero entries A similar
argument shows that each column has only a nite numb er of non zero entries
Now for given q consider the rows of the nite matrix B indexed by con gurations of
typ e q and such that the tap e head is lo cated at one of the k non b order cells Then
Consider for example the space of nite complex linear combinations of p ositiveintegers and the linear op erator which
maps jii to ji i
k
jT j k card Eachrowof B indexed by a con guration c T has the prop erty that if d is any
con guration that yields c with non zero amplitude in a single step then d S Therefore the row indexed
by c in the nite matrix B has the same non zero entries as the row indexed by c in the in nite matrix
Therefore it makes sense to say that row c of B is of the same typ e as row c of the in nite matrix Finally
the rows of eachtyp e constitute a fraction at least jT j m of all rows of B Substituting the b ounds from
k
k card
ab ove we get that this fraction is at least Since k this is at least
k
a
k card card Q k card
for constant a card card Q card This establishes all the prop erties of the matrix B
used in the pro of ab ove
B Reversible TMs are as p owerful as deterministicTMs
In this app endix weprove the Synchronization Theorem of x
We b egin with a few simple facts ab out reversible TMs We give necessary and su cient conditions
for a deterministic TM to b e reversible and weshow that just as for QTMs a partially de ned reversible
TM can always b e completed to giveawell formed reversible TM Wealsogive as an aside an easy pro of
that reversible TMs can e ciently simulate reversible generalized TMs
Theorem B ATMorgeneralizedTM M is reversible i the fol lowing two conditions both hold
Each state of M can beentered while moving in only one direction In other words if p
q d and p q d then d d
The transition function is one to one when direction is ignored
Pro of First weshow that these two conditions imply reversibility
Supp ose M Q is a TM or generalized TM satisfying these two conditions Then the following
pro cedure lets us takeany con guration of M and compute its predecessor if it has one First since each
state can b e entered while moving in only one direction the state of the con guration tells us in whichcell
the tap e head must have b een in the previous con guration Lo oking at this cell we can see what tap e
symbol was written in the last step Then since is one to one weknow the up date rule if any that was
used on the previous step allowing us to reconstruct the previous con guration
Next we show that the rst prop ertyisnecessaryforreversibility So for example consider a TM
or generalized TM M Q such that p q L and p q R Then we can
easily construct two con gurations which lead to the same next con guration Let c be an y con guration
where the machine is in state p reading a and where the symbol two cells to the left of the tap e head
is a and let c b e identical to c except that the and are changed to and the machine
is in state p and the tap e head is two cells further left Therefore M is not reversible Since similar
arguments apply for each pair of distinct directions the rst condition in the theorem must b e necessary
for reversibility
Finally we show that the second condition is also necessary for reversibility Supp ose that M Q
is a TM or generalized TM with p p Then any pair of con gurations which di er only in
the state and symb ol under the tap e head where one has p and the other p lead to the same
next con guration and again M is not reversible
Corollary B If M is a reversible TM then every con guration of M has exactly one predecessor
Pro of Let M Q bea reversible TM By the de nition of reversibility each con guration of M
has at most one predecessor
So let c b e a con guration of M in state q Theorem B tells us that M can enter state q while
moving its tap e head in only one direction d Since Theorem B tells us that ignoring direction is
q
one to one taking the inverse of on the state q and the symb ol in direction d tells us how to transform
q
c into its predecessor
Corollary B If isapartial function from Q to Q fL Rg satisfying the two conditions
of Theorem B then can beextended to a total function that stil l satis es Theorem B
Pro of Supp ose is a partial function from Q to Q fL R g that satis es the prop erties of
Theorem B Then for each q Q let d b e the one direction if any in which q can b e entered and
q
let d b e arbitrarily L otherwise Then we can ll in unde ned values of with as yet unused triples of
q
the form q d so as to maintain the conditions of Theorem B Since there are card card Q
q
such triples there will b e exactly enough to fully de ne
Theorem B If M is a generalizedreversible TM then thereisareversible TM M that simulates
M with slowdown at most
Pro of The idea is to replace any transition that has the tap e head stand still with two transitions The
rst up dates the tap e and moves to the right rememb ering which state it should enter The second steps
back to the left and enters the desired state
So if M Q is a generalized reversible TM then we let M b e identical to M except that for
each state q with a transition of the form p q N we add a new state q and we also add a new
q q L for each Finally we replace each transition p q N with transition rule
the transition p q R Clearly M simulates M with slowdown by a factor of at most
To complete the pro of weneedtoshow that M is also reversible
So consider a con guration c of M We need to show that c has at most one predecessor If c is in
state q Q and M enters q moving left or right then the transitions into q in M are identical to those
in M and therefore since M is reversible c has at most one predecessor in M Similarly ifc is in one of
the new states q then the transitions into q in M are exactly the same as those into q in M except that
the tap e head moves right instead of staying still So again the reversibilityofM implies that c has at
most one predecessor Finally supp ose c is in state q Q where M enters q while standing still Then
Theorem B tells us that all transitions in M that enter q have direction N Therefore all of them have
b een removed and the only transitions entering q in M are the new ones of the form q q L
Again this means that c can have only one predecessor
We will prove the Synchronization Theorem in the following way Using ideas from the constructions
of Bennett and Morita et al wewillshow that given any deterministic TM M there is a reversible
multi trackTMwhich on input x pro duces output x M x and whose running time dep ends only on the
sequence of head movements of M on input x Then since any deterministic computation can b e simulated
e ciently by an oblivious machine whose head movements dep end only on the length of its input we
will b e able to construct the desired synchronized reversible TM
The idea of Bennett s simulation is to run the target machine keeping a history to maintain reversibility
Then the output can b e copied and the simulation run backwards so that the history is exactly erased
while the input is recovered Since the target tap e head moves back and forth while the history steadily
grows Bennett uses a multi tap e TM for the simulation
The idea of Morita et al s simulation is to use a simulation tap e with several tracks some of whichare
used to simulate the tap e of the desired machine and some of which are used to keep the history Provided
that the machine can movereversibly b etween the current head p osition of the target machine and the end
of the history it can carry out Bennett s simulation with a quadratic slowdown Morita et al work with
TMs with one way in nite tap es so that the simulating machine can movebetween the simulation and the
history bymoving left to the end of the tap e and then searching backtowards the right In our simulation
we write down history for every step the target machine takes rather than just the non reversible steps
This means that the end of the history will always b e further right than the simulation allowing us to work
with two way in nite tap es Also we use reversible TMs that can only mo ve their head in the directions
fL Rg rather than the generalized reversible TMs used by Bennett and Morita et al
De nition B A deterministic TM is oblivious if its running time and the position of its tapehead
at eachtimestepdepend only on the length of its input
In carrying out our single tap e Bennett constructions we will nd it useful to rst build simple reversible
TMs to copy a string from one track to another and to exchange the strings on two tracks We will copy
strings delimited by blanks in the single tap e Bennett construction but will copy strings with other
delimiters in a later section
Lemma B For any alphabet there is a normal form reversible TM M with alphabet with
the fol lowing property When run on input x yM runs for max jxj jy j steps returns the tapehead
to the start cel l and outputs y x
Pro of We let M have alphab et state set fq q q q q g and transition function de ned by
f
other
q q L q L
q q R
q q L q R
q q R q L
f
q q R q R
f
Since each state in M can b e entered in only one direction and its transition function is one to one
M is reversible Also it can b e veri ed that M p erforms the desired computation in the stated number of
steps
Lemma B For any alphabet there is a normal form reversible TM M with alphabet with
the fol lowing property When run on input x M outputs x x and when run on input x x it outputs x In
either case M runs for jxj steps and leaves the tapehead back in the start cel l
Pro of We let M have alphab et state set fq q q q q g and transition function de ned bythe
f
following where each transition is duplicated for each non blank
q q L q L q L
q R q
q q L q R q R
q R q L q L q
f
q q R q R q R
f
Since each state in M can b e entered in only one direction and its transition function is one to one
M can b e extended to a reversible TM Also it can b e veri ed that M p erforms the desired computation
in the stated numb er of steps
Theorem B Let M be an oblivious deterministic TM which on any input x produces output M x
with no embedded blanks with its tapehead back in the start cel l and with M x beginning in the start
cel l Then thereisareversible TM M which on input x produces output x M x and on input x M x
produces output x Inboth cases M halts with its tapehead back in the start cel l and takes time which
depends only on the lengths of x and M x and which is bounded by a quadratic polynomial in the running
time of M on input x
Pro of Let M Q b e an oblivious deterministic TM as stated in the theorem and let q q be the
f
initial and nal states of M
The simulation will run in three stages
M will simulate M maintaining a history to make the simulation reversible
M will copy its rst track as shown in Lemma B
M runs the reverse of the simulation of M erasing the history while restoring the input
We will construct a normal form reversible TM for each of the three stages and then dovetail them
together
Each of our machines will b e a four track TM with the same alphab et The rst track with alphab et
will b e used to simulate the tap e of M The second track with alphab et f g will b e used
to store a single lo cating the tap e head of M The third track with alphab et f g Q
will b e used to write down a list of the transitions taken by M starting with the marker This will
help us nd the start cell when weenter and leave the copying phase The fourth track with alphab et
will b e used to write the output of M
In describing the rst machine we will give a partial list of transitions ob eying the conditions of
Theorem B and then app eal to Corollary B Our machines will usually only b e reading and
writing one track at a time So for convenience we will list a transition with one or more o ccurrences of
the symb ols of the form v and v to stand for all p ossible transitions with v replaced by a symb ol from
i i
i
replaced byasymbolfrom other than the appropriate other than the sp ecial marker and with v
i i
i
and
The rst phase of the simulation will b e handled by the machine M with state set given by the union
of sets Q Q Q Q and fq q g Its start state will b e q and the nal state q
a b a f
The transitions of M are de ned as follows First we mark the p osition of the tap e head of M in the
start cell mark the start of the history in cell and enter state q
q v v v v q R
a b
q v v v v q R
b
Then for eachpairp with p q and with transition p q d in M we include transitions
f
to go from state p to state q up dating the simulated tap e of M while adding p to the end of the history
W e rst carry out the up date of the simulated tap e rememb ering the transition taken and then moveto
the end of the history to dep osit the information on the transition If we are in the middle of the history
we reach the end of the history bywalking rightuntil we hit a blank However if we are to the left of the
history then wemust rst walk rightover blanks until wereach the start of the history
p v v v v q p d
q p v v v v q p R
q p v v v v v v q p R
q p v v v v q p R
q p v v v v q p R
v v v v q p R q p
q p v v v v v v q p R
q p v v v p v q R
hine reaches state q it is standing on the rst blank after the end of the history So When the mac
for each state q Q we include transitions to move from the end of the history back left to the head
p osition of M Weenter our walk left state q while writing a on the history tap e So to maintain
reversibility wemust enter a second left walking state q to lo ok for the tap e head marker past the left
end of the history When we reach the tap e head p osition we step left and rightentering state q
q v v v v q L
q L q v v v v v v
q v v v v q L
q v v v v v v q L
q v v v v q L
q v v v v q L
q v v v v q L
q v v v v v v v v q R
Finally we put M in normal form by including the transition
q q R
f a
for each in the simulation alphab et
It can b e veri ed that each state can b e entered in only one direction using the ab ove transitions and
relying on the fact that M is in normal form and we don t simulate it s transitions from q backto q it
f
can also b e veri ed that the partial transition function describ ed is one to one Therefore Corollary B
says that we can extend these transitions to give a reversible TM M Notice also that the op eration of
M is indep endent of the contents of the fourth track and that it leaves these contents unaltered
For the second and third machines we simply use the copying machine constructed in Lemma B
ab ove and the reverse of M constructed using Lemma on page Since M op erated indep endently
of the fourth track so will its reversal Therefore dovetailing these three TMs gives a reversible TM M
whichoninputx pro duces output x M x with its tap e head back in the start cell and on input
x M x pro duces output x
Notice that the time required by M to simulate a step of M is b ounded by a p olynomial in the running
time of M and dep ends only on the current head p osition of M the direction M moves in the step and
how many steps have already b een carried out Therefore the running time of the rst phase of M de
p ends only on the series of head movements of M In fact since M is oblivious this running time dep ends
only the length of x The same is true of the third phase of M since the reversal of M takes exactly two
extra time steps Finally the running time of the copying machine dep ends only on the length of M x
Therefore the running time of M on input x dep ends only on the lengths of x and M x and is b ounded
by a quadratic p olynomial in the running time of M on x
Theorem B Let M be an oblivious deterministic TM which on any input x produces output M x
with no embedded blanks with its tapehead back in the start cel l with M x beginning in the start cel l
and such that the length of M x depends only on the length of x Let M be an oblivious deterministic
TM with the same alphabet as M which on any input M x produces output x with its tapehead back in
the start cel l and with x beginning in the start cel l Then thereisareversible TM M which on input x
produces output M x Moreover M on input x halts with its tapehead back in the start cel l and takes
time which depends only on the length of x and which is boundedbyapolynomial in the running time of
M on input x and the running time of M on M x
Pro of Let M and M b e as stated in the theorem with the common alphab et
Then the idea to construct the desired M is rst to run M to compute x M x then to run an
exchange routine to pro duce M x x and nally to run M to erase the string x where each of the three
phases starts and ends with the tap e head in the start cell Using the construction in Theorem B we
build normal form reversible TMs to accomplish the rst and third phases in times which dep end only on
the lengths of x and M x and b ounded by a p olynomial in the running times of M on input x and M
on input M x In Lemma B on page wehave already constructed a reversible TM that p erforms
hange in time dep ending only on the lengths of the two strings Dovetailing these three machines the exc
gives the desired M
Since any function computable in deterministic p olynomial time can b e computed in p olynomial time
by an oblivious generalized deterministic TM Theorems B and B together with Theorem
give us the following
Theorem Synchronization Theorem If f is a function mapping strings to strings which
can becomputed in deterministic polynomial time and such that the length of f x depends only on the
length of x then thereisapolynomial time stationary normal form QTM which given input x produces
output x f x and whose running time depends only on the length of x
If f is a function from strings to strings that such that both f and f can becomputed in deterministic
polynomial time and such that the length of f x depends only on the length of x then thereisapolynomial
time stationary normal form QTM which given input x produces output f x and whose running time
depends only on the length of x
C Areversible lo oping TM
We prove here the Lo oping Lemma from x
Lemma Lo oping Lemma There is a stationary normal form reversible TM M and a
constant c with the fol lowing properties On input any positive integer k written in binary M runs for
c
time O k log k and halts with its tape unchanged Moreover M has a special state q such that on input
k M visits state q exactly k times each time with its tapehead back in the start cel l
Pro of As mentioned ab ovein x the di culty is to construct a lo op with a reversible entrance and exit
We accomplish this as follows Using the Synchronization Theorem we can build three track stationary
normal form reversible TM M Q running in time p olynomial in log k that on input b x k where
b f g outputs b x k where b is the opp osite of b if x ork but not b oth and b b
otherwise Calling the initial and nal states of this machine q q we construct a reversible M that
f
lo ops on machine M as follows We will give M new initial and nal states q q and ensure that it has
a z
the following three prop erties
Started in state q with a on the rst track M steps left and backright changing the to a
a
and entering state q
When in state q witha onthe rsttrack M steps left and backrightinto state q
f
When in state q with a on the rst track M steps left and backright changing the to a and
f
halts
So on input kM will step left and rightinto state q c hanging the tap e contents to k Then
machine M will run for the rst time changing k to k and halting in state q Whenever M is
f
in state q with a on the rst track it reenters q to run M again So machine M will run k more
f
times until it nally pro duces k k At that p oint M changes the tap e contents to k k and halts So
on input k M visits state q exactly k times each time with its tap e head back in the start cell This
f
means we can construct the desired M by identifying q as q and dovetailing M b efore and after with
f
reversible TMs constructed using the Synchronization Theorem to transform k to k and k k back
to k
We complete the pro of by constructing a normal form reversible M that satis es the three prop erties
ab ove We give M the same alphab et as M and additional states q q q q The transition function
a b y z
for M is the same as that of M for states in Q q and otherwise dep ends only on the rst track leaving
f
the others unchanged and is given by the following table
q q L
a b
q R q
b
q L q L q
b y f
q q R
y z
q q R q R q R
z a a a
It is easy to see that M is is normal form and satis es the three prop erties stated ab ove Moreover since
M is reversible and ob eys the two conditions of Theorem B it can b e veri ed that the transition
function of M also ob eys the two conditions of Theorem B Therefore according to Theorem B
the transition function of M can b e completed giving a reversible TM
Acknowledgements
This pap er has greatly b ene tted from the careful reading and many useful comments of Richard Jozsa
and Bob Solovay We wish to thank them as well as Gilles Brassard Charles Bennett Noam Nisan and
Dan Simon
References
Adleman L DeMarrais J and Huang M Quantum computability manuscript
Arora S Impagliazzo R Vazirani U On the Role of the Cook Levin Theorem in Complexity
Theory manuscript
Arora S Lund C Motwani R Sudan M and SzegedyM Proof veri cation and intractabil
ity of approximation problems Pro ceedings of the rd Annual IEEE Symp osium on Foundations of
Computer Science
Babai L and Moran S Arthur Merlin games arandomizedproof system and a hierarchy of
complexity classes Journal of Computer and System Sciences pp
Barenco A Bennett C Cleve R DiVincenzo D Margolus N Shor P
Sleator T Smolin J and Weinfurter H Elementary gates for quantum computation
manuscript
Benioff P Quantum Hamiltonian models of Turing machines Journal of Statistical Physics
pp
Bennett C H Logical reversibility of computation IBM J Res Develop pp
Bennett C H Time spacetradeo s for reversible computation SIAM J Comput
pp
Bennett C H Bernstein E Brassard G and Vazirani U Strengths and weaknesses of
quantum computing SIAM J Comput to app ear
Bernstein E Quantum complexity theory PhD dissertation U C Berkeley to app ear
Bernstein E and Vazirani U Quantum complexity theory Pro ceedings of the th Annual
ACM Symp osium on Theory of Computing pp
Berthiaume A and Brassard G The quantum chal lenge to structural complexity theory Pro
ceedings of th IEEE Conference on Structure in Complexity Theory pt
Berthiaume A and Brassard G Oracle quantum computing Journal of Mo dern Optics
pp
Berthiaume A Deutsch D and Jozsa R The stabilisation of quantum computation Pro
ceedings of the Workshop on Physics and Computation p
Bshouty N and Jackson J Le arning DNF over uniform distribution using a quantum example
oracle Pro ceedings of th Annual ACM Conference on Computational Learning Theory pp
Calderbank A R and Shor P Good quantum error correcting codes exist Phys Rev A to
app ear
Chirac and Zoller Quantum computation using trappedcold ions manuscript
Chuang I LaFlamme R Shor P and Zurek W Quantum computers factoring and
decoherence manuscript
Cohen Tannoudji C Diu B LaLoe F Quantum mechanics pp
Deutsch D Quantum theory the Church Turing principle and the universal quantum computer
Pro c R So c Lond A pp
Deutsch D Quantum computational networks Pro c R So c Lond A pp
Deutsch D and Jozsa R Rapid solution of problems by quantum computation Pro c R So c
Lond A pp
DiVincenzo D Two bit gates are universal for quantum computation Phys Rev A
pp
Durr C Santha M Thanh Adecision procedure for wel l formed linear quantum cel lular au
tomata RACworkshop Berkeley
Feynman R Simulating physics with computers International Journal of Theoretical Physics
pp
Feynman R Quantum mechanical computers Foundations of Physics pp Orig
inally app eared in Optics News February
Fredkin E and Toffoli T Conservative Logic Internat J Theoret Phys p
Grover L Searching for a need le in a haystack a fast quantum mechanical algorithm manuscript
Landauer R Is quantum mechanics useful Pro c R So c Lond to app ear
Lipton R Personal communication
Lloyd S Apotential ly realizable quantum computer Science pp
Machta J Phase information in quantum oracle computing Physics Department Universityof
Massachusetts at Amherst manuscript
Morita K Shirasaki A and Gono Y A tape symbol reversible Turing Machine IEEE
Transactions of the IEICE E pp
von Neumann J Various Techniques UsedinConnection with Random Digits Notes byG E
Forsythe National Bureau of Standards Applied Math Series pp Reprinted in von
Neumann s Collected Works Pergamon Press pp
Palma G Suominen K and Ekert A Quantum computers and dissipation manuscript
A Shamir IP PSPACE Pro ceedings of the nd ACM Symp osium on the Theory of Computing
pp
Shor P Algorithms for quantum computation Discrete log and factoring Pro ceedings of the th
Annual IEEE Symp osium on Foundations of Computer Science
Shor P Fault tolerant quantum computation Pro ceedings of the th Annual IEEE Symp osium on
the Foundations of Computer Science
Simon D On the power of quantum computation Pro ceedings of the th Annual IEEE Symp osium
on Foundations of Computer Science
Solovay R and Yao A manuscript
Toffoli T Bicontinuous extensions of invertible combinatorial functions Mathematical Systems
Theory pp
Unruh W Maintaining coherenceinquantumcomputers Phys Rev A p
Valiant L Personal communication
Vazirani U and Vazirani V Random polynomial time is equal to semi random polynomial time
Pro ceedings of the th Annual IEEE Symp osium on Foundations of Computer Science
Watrous J On one dimensional quantum cel lular automata Pro ceedings of the th Annual IEEE
Symp osium on Foundations of Computer Science
Yao A Quantum circuit complexity Pro ceedings of the th Annual IEEE Symp osium on Founda
tions of Computer Science pp
Zuckerman D Weak Random Sources PhD dissertation U C Berkeley