arXiv:quant-ph/0605030v3 11 Aug 2007 TOGYUIESLQATMTRN AHNSADINVARIANCE AND MACHINES TURING QUANTUM UNIVERSAL STRONGLY lxt,UieslQatmCmue,QatmKolmogorov Quantum Problem. Computer, Halting Complexity, Quantum Universal plexity, a lasb fetvl eopsdit lsia n a and classical a into strin decomposed qubit effectively input part. an be spac quantum in halting encoded always information orthogonal can the mutually that behavi of show halting notion and their the of introduce analysis We thorough u a only including UQTM QTMs, the of choice constant. et the additive on Berthiaume an depends by to i.e. defined invariant, is as for al. QTM complexity other Kolmogorov steps. every quantum time simulate of can shown number have preassigned that but who UQTM arbitrary, Vazirani a an and is Bernstein there by that QTM work other extends every This halted simulate has QTM can other that the (UQTM) machine ing eln(-al [email protected]). (e-mail: Berlin navne hc sdfcl oaheei h unu case. quantum the in achieve computation to the difficult of steps is time which does of advance, paper number in the this quantum know th in to is constructed need study not difference QTM to universal” basic “strongly needed The differen the is theory. a is information which This algorithmic universality, lengths. program of of notion sense the in universal the especially complexity, BQP. computational class complexity quantum foundati study the provides to result every that simulates remarkable QTM efficiently This a it QTM. exists the that possible sense there out the that in worked proved universal and [3] is quantum idea detail Vazirani a more earlier and in even of Bernstein theory an [2]. model on Feynman first In elaborating the by the realm. (QTM), generalize proposed quantum machine to Turing the [1] done to Deutsch been computation 1985, universal has of work concept much years, recent respectively. algorith and theory complexity computi information computational general in possible as every study once, at for to device valid opportunity computation the of techno properties provided computer it of modern Moreover, notion of simu- ogy. This foundation can machine. the that computing laid universality possible (TM), other machine every Turing late universal the device, O .Mulri ihteIsiueo ahmtc,TcnclU Technical Mathematics, of Institute the with is M¨uller M. togyUieslQatmTrn ahnsand Machines Turing Quantum Universal Strongly ne Terms Index for framework mathematical new a on based is that proof proof Our rigorous a give we result, this to corollary a As Abstract nti ae,w hl hwta hr xssaQMta is that QTM a exists there that show shall we paper, this In in theory information quantum of development the to Due newsteisgtta hr sasnl computing single a is there that sci- insight computer the of was breakthroughs ence fundamental the of NE W hwta hr xssauieslqatmTur- quantum universal a exists there that show —We QatmTrn ahn,Klooo Com- Kolmogorov Machine, Turing —Quantum naineo omgrvComplexity Kolmogorov of Invariance .I I. n hnhl tefwt rbblt one. probability with itself halt then and NTRODUCTION iest of niversity aksM¨uller Markus other until our. mic on ng es at l- p g FKLOOO OPEIY(UUT1,20)1 2007) 11, (AUGUST COMPLEXITY KOLMOGOROV OF t usinwe ecncnie T shaving as QTM a consider can we to. when pointing is question head the where cell tape single amplitude the transition local a by determined aecls h T sawoeeovsuiaiyi discrete in the unitarily along evolution evolves time moves whole unitary that (global) a The head as steps. tape time QTM single The a cells. and tape control, a tape, .QatmTrn ahnsadterHligConditions Halting their and Machines Turing Quantum A. amplitude on transition based local has a is of [6] which instead definition Benioff Hamiltonian different local evolution. slightly a time transition a QTM’s unitary out worked a in that gave rele- results who conditions Additional function [5], sufficient [4]. Nishimura and and Gruska necessary Ozawa by includes book literature the vant see Vazirani, and iial oacasclTM classical II Subsection a in to detail in Similarly model their describe We Vazirani. rhgnlt h atn tt taytime any at state halting the to orthogonal o prilrcriefnto from function recursive “partial for ecnrprs hsdfiiinas definition this rephrase can We input atn time. halting uscinI-) hr saspecified a is space there Hilbert II-B), finite-dimensional Subsection the on state H mixed some is { up( supp o otiptqbtstrings qubit input most for codn o[] esyta h QTM the that say we [3], to According ento sueu n aua,a es o h ups to purpose the this for that least below at argue complexity. We natural, Kolmogorov paper. quantum and [3], study this useful Vazirani in is and use t get definition Bernstein detail also not by in we will analyze halting time We which rather for [11]. unitary but definition [10], discussion, with simple this [9], compatible into [8], deep is [7], too this e.g. how see evolution, and input some ..tecnrlis control the i.e. n ilb oeabtaynumber arbitrary some be will nsm tt otie ypriltaeoe h l h othe the all denote the we over which trace QTM) partial the of by parts (obtained state some in λ, h 1 C q hr a enavvddsuso nteltrtr nthe on literature the in discussion vivid a been has There o opc rsnaino h eut yBernstein by results the of presentation compact a For u icsinwl eyo h ento yBrsenand Bernstein by definition the on rely will discussion Our ngnrl h vra of overlap the general, In ups QTM a Suppose uisfor qubits euetetrs“uigmcie T)ad“optr synony “computer” and (TM) machine” “Turing terms the use We 0 f , | htdsrbstecnrl ydfiiino T (see QTM a of definition By control. the describes that 1 M | , M ψ 00 C T i C t . . . , ( if | ( ψ | ψ } i t ) i eoe h nt iaystrings. binary finite the denotes | )) ieses h control The steps. time q f i | ⊥ 1 = exactly M q and f uso oeqatminput quantum some on runs i ..tecnrlsaei exactly is state control the i.e. , ntefia tt ttime at state final the in 1 h M | ψ T ossso ninfinite an of consists QTM a , q { f i 0 C t between | hr ilb otime no be will there , , M ( 1 | } ψ C ∗ t i M ( ) to | ψ M ihtefia state final the with C nlstate final t { C i 0 M eoadoe Hence, one. and zero ( ) C T , | | δ ψ 1 q ( of } f | at ttime at halts i hc nyaffects only which ψ ∗ T < t ) i ,where ”, ngnrl this general, In . M i U 0 = = ) scompletely is ilte be then will | q f eoethe before halted ∀ H ∈ i { | q T. < t 0 T f , T | mously ψ 1 ih and , T } i ∈ | q ∗ q -B. . f on on he C f of N = | i r . , STRONGLY UNIVERSAL QUANTUM TURING MACHINES AND INVARIANCE OF KOLMOGOROV COMPLEXITY (AUGUST 11, 2007) 2 such that the aforementioned halting conditions are satisfied. are quantum operations, which is a very useful and plausible We call those qubit strings non-halting, and otherwise T - property. halting, where T N is the corresponding halting time. Even more important, at each single time step, an out- In Subsection∈ III-A, we analyze the resulting geometric side observer can make a measurement of the control state, structure of the halting input qubit strings. We show that described by the operators q q and 1 q q (thus | f ih f | − | f ih f | inputs ψ with some fixed length n that make the QTM M observing the halting time), without spoiling the computation, | i (n) halt after t steps form a linear subspace (t). Moreover, as long as the input ψ is halting. As soon as halting is M | i inputs with different halting times are mutuallyH orthogonal, detected, the observer can extract the output quantum state (n) (n) from the output track (tape) and use it for further quantum i.e. M (t) M (t′) if t = t′. According to the halting conditionsH given⊥ H above, this is6 almost obvious: Superpositions information processing. This is true even if the halting time of t-halting inputs are again t-halting, and inputs with different is very large, which typically happens in the study of Kol- halting times can be perfectly distinguished, just by observing mogorov complexity. Consequently, our definition of halting their halting time. has the useful property that if an outside observer is given In Figure 1, a geometrical picture of the halting space some unknown quantum state ψ which is halting, then the | i structure is shown: The whole space R3 represents the space observer can find out with certainty by measurement. of inputs of some fixed length n, while the plane and the Finally, if we instead introduced some probabilistic notion (n) of halting (say, we demanded that we observe halting of straight line represent two different halting spaces M (t′) (n) H the QTM M at some time t with some large probability and M (t). Every vector within these subspaces is perfectly halting,H while every vector “in between” is non-halting and p < 1), then it would not be so clear how to define not considered a useful input for the QTM M. quantum Kolmogorov complexity correctly. Namely if the halting probability is much less than one, it seems necessary to introduce some kind of “penalty term” into the definition of quantum Kolmogorov complexity: there should be some trade-off between program length and halting accuracy, and it is not so clear what the correct trade-off should be. For example, what is the complexity of a qubit string that has a program of length 100 which halts with probability 0.8, and another program of length 120 which halts with probability 0.9? The definition of halting that we use in this paper avoids such questions.
Fig. 1. mutually orthogonal halting spaces. B. Different Notions of Universality for QTMs Bernstein and Vazirani [3] have shown that there exists a universal QTM (UQTM) . It is important to understand At first, it seems that the halting conditions given above are what exactly they mean by “universal”.U According to [3, Thm. far too restrictive. Don’t we loose a lot by dismissing every 7.0.2], this UQTM has the property that for every QTM M input which does not satisfy those conditions perfectly, but, there is some classicalU bit string s 0, 1 (containing a say, only approximately up to some small ε? To see that it is M ∗ description of the QTM M) such that∈ { } not that bad, note that T (1) most (if not all) of the well-known quantum algorithms, (sM , T, δ, ψ ) MO( ψ ) Tr <δ • U | i − R | i like the quantum Fourier transform or Shor’s algorithm, for every input ψ , accuracy δ > 0 and number of time steps have classically controlled halting. That is, the halting | i T T N. Here, Tr is the trace distance, and MO( ψ ) time is known in advance, and can be controlled by a is∈ the contentk·k of the output tape O of M afterR T steps| i of classical subprogram. computation (the notation will be defined exactly in Subsec- we show elsewehere [12] (cf. Theorem 3.15) that every • tion II-B). input that is almost halting can be modified by adding This means that the UQTM simulates every other QTM at most a constant number of qubits to halt perfectly, i.e. M within any desired accuracyU and outputs an approximation to satisfy the aforementioned halting conditions. This can of the output track content of M and halts, as long as the be interpreted as some kind of “stability result”, showing number of time steps T is given as input in advance. that the halting conditions are not “unphysical”, but have Since the purpose of Bernstein and Vazirani’s work was some kind of built-in error tolerance that was not expected to study the computational complexity of QTMs, it was a from the beginning. reasonable assumption that the halting time T is known in Moreover, this definition of halting is very useful. Given advance (and not too large) and can be specified as additional two QTMs M1 and M2, it enables us to construct a QTM M input. The most important point for them was not to have short which carries out the computations of M1, followed by the inputs, but to prove that the simulation of M by is efficient, computations of M , just by redirecting the final state q of i.e. has only polynomial slowdown. U 2 | f i M1 to the starting state q0 of M2 (see [3, Dovetailing Lemma The situation is different if one is interested in studying 4.2.6]). In addition, it follows| i from this definition that QTMs quantum algorithmic information theory instead. It will be STRONGLY UNIVERSAL QUANTUM TURING MACHINES AND INVARIANCE OF KOLMOGOROV COMPLEXITY (AUGUST 11, 2007) 3 explained in Subsection I-C below that the universality notion basis for the whole theory of Kolmogorov complexity (see (1) is not enough for proving the important invariance property [13]). Basically, it states that the choice of the computer M is of quantum Kolmogorov complexity, which says that quantum not important as long as M is universal; choosing a different Kolmogorov complexity depends on the choice of the universal universal computer will alter the complexity only up to some QTM only up to an additive constant. additive constant. More specifically, there exists a universal To prove the invariance property, one needs a generalization computer U such that for every computer M there is a constant of (1), where the requirement to have the running time T as cM N such that additional input is dropped. We show below in Section III ∈ C (s) C (s)+ c for every s 0, 1 ∗. (2) that there exists a UQTM U that satisfies such a generalized U ≤ M M ∈{ } universality property, i.e. that simulates every other QTM until This so-called “invariance property” follows easily from the that other QTM has halted, without knowing that halting time existence of a universal computer U in the following sense: in advance, and then halts itself. There exists a computer U such that for every computer M Why is that so difficult to prove? At first, it seems that one and every input s 0, 1 ∗ there is an input s˜ 0, 1 ∗ such can just program the UQTM mentioned in (1) to simulate that U(˜s)= M(s)∈{and ℓ(˜}s) ℓ(s)+ c , where∈{c }N is a U M M the other QTM M for T = 1, 2, 3,... time steps, and, after constant depending only on ≤M. In short, there is a computer∈ every time step, to check if the simulation of M has halted U that produces every output that is produced by any other or not. If it has halted, then halts itself and prints out the computer, while the length of the corresponding input blows U output of M, otherwise it continues. up only by a constant summand. One can think of the bit string This approach works for classical TMs, but for QTMs, there s˜ as consisting of the original bit string s and of a description is one problem: in general, the UQTM can simulate M of the computer M (of length c ). U M only approximately. The reason is the same as for the circuit The quantum generalization of Kolmogorov complexity that model, i.e. the set of basic unitary transformations that can we consider in this paper has been first defined by Berthiaume, U apply on its tape may be algebraically independent from that van Dam and Laplante [14]. Basically, they define the quantum of M, making a perfect simulation in principle impossible. Kolmogorov complexity QC of a string of qubits ψ as the But if the simulation is only approximate, then the control length of the shortest string of qubits that, when given| i as input state of M will also be simulated only approximately, which to a QTM M, makes M output ψ and halt. (We give a will force to halt only approximately. Thus, the restrictive formal definition of a “qubit string”| ini Subsection II-A and of U halting conditions given above in Equation (6) will inevitably quantum Kolmogorov complexity QC in Subsection II-C). be violated, and the computation of will be treated as invalid In [14], it is claimed that quantum Kolmogorov complexity U and be dismissed by definition. QC is invariant up to an additive constant similar to (2). It is This is a severe problem that cannot be circumvented easily. stated there that the existence of a universal QTM in the Many ideas for simple solutions must fail, for example the idea sense of Bernstein and Vazirani (see Equation (1))U makes it to let compute an upper bound on the halting time T of all possible to mimic the classical proof and to conclude that the U inputs for M of some length n and just to proceed for T UQTM outputs all that every other QTM outputs, implying time steps: upper bounds on halting times are not computable. invarianceU of quantum Kolmogorov complexity. Another idea is that the computation of should somehow But this conclusion cannot be drawn so easily, because (1) U consist of a classical part that controls the computation and demands that the halting time T is specified as additional a quantum part that does the unitary transformations on the input, which can enlarge the input length dramatically, if data. But this idea is difficult to formalize. Even for classical T is very large (which typically happens in the study of TMs, there is no general way to split the computation into Kolmogorov complexity). “program” and “data” except for special cases, and for QTMs, As explained above in Subsection I-B, it is not so easy to by definition, global unitary time evolution can entangle every get rid of the halting time. The main reason is that the UQTM part of a QTM with every other part. can simulate other QTMs only approximately. Thus, it will Our proof idea rests instead on the observation that every alsoU simulate the control state and the signaling of halting only input for a QTM which is halting can be decomposed into a approximately, and cannot just “halt whenever the simulation classical and a quantum part, which is related to the mutual has halted”, because then, it will violate the restrictive halting orthogonality of the halting spaces. See Subsection I-E for conditions given in Equation (6). As we have chosen this details. definition of halting for good reasons (cf. the discussion at the beginning of Subsection I-A above), we do not want to C. Q-Kolmogorov Complexity and its Supposed Invariance drop it. The classical Kolmogorov complexity CU (s) of a finite bit Instead of (1), a stronger notion of universality is needed, U string s 0, 1 ∗ is defined as the minimal length of any namely a “strongly universal” QTM that, as explained computer∈ program { } p that, given as input into a TM M, outputs above in Subsection I-B, simulates every other QTM M the string and makes M halt: until the other QTM has halted and then halts itself with probability one, as required by the halting conditions given C (s) := min ℓ(p) M(p)= s . M { | } in Subsection I-A. Then, the classical proof outlined above For this quantity, running times are not important; all that can be carried over to the quantum situation. In this paper, we matters is the input length. There is a crucial result that is the prove that such a QTM U really exists (Theorem 1.1), and as STRONGLY UNIVERSAL QUANTUM TURING MACHINES AND INVARIANCE OF KOLMOGOROV COMPLEXITY (AUGUST 11, 2007) 4
(n) a corollary, the invariance property for quantum Kolmogorov M (τ) that contains ϕ is only a subspace and might have complexity follows (Theorem 1.2). Hmuch smaller dimension| dTuring Machine): might be huge and may not have a short description. Instead, There is a fixed-length quantum Turing machine U such that we can encode the halting number. Define the halting time for every QTM M and every qubit string σ for which M(σ) N sequence ti i=1 as the set of all integers t N such that is defined, there is a qubit string σM such that (n) { } ∈ dim M (t) 1, ordered such that ti < ti+1 for every i, U (σ ,δ) M(σ) <δ thatH is, the set≥ of all halting times that can occur on inputs of k M − kTr length n. Thus, there must be some i N such that τ = t , for every δ Q+, where the length of σ is bounded by ∈ i ∈ M and i can be called the halting number of ϕ . Now, we assign ℓ(σM ) ℓ(σ)+ cM , and cM N is a constant depending | i ≤ ∈ code words ci to the halting numbers i, that is, we construct only on M. N a prefix code ci i=1 0, 1 ∗. We want the code words to Note that σM does not depend on δ. We conclude from be short; we claim{ } that⊂{ we can} always choose the lengths as this theorem and a two-parameter generalization given in (n) Proposition 3.14 that quantum Kolmogorov complexity as ℓ(ci)= n +1 log2 dim M (ti) . defined in [14] is indeed invariant, i.e. depends on the choice −⌈ H ⌉ of the strongly universal QTM only up to some constant: This can be verified by checking the Kraft inequality: Theorem 1.2 (Invariance of Q-Kolmogorov Complexity): N N (n) ℓ(ci) n log dim (ti) 1 There is a fixed-length quantum Turing machine U such that 2− = 2− 2⌈ 2 HM ⌉− for every QTM M there is a constant cM N such that i=1 i=1 ∈ X Xn n (n) n 2 n QCU(ρ) QCM (ρ)+ cM for every qubit string ρ. 2− dim (t ) 2− dim C ⊗ ≤ ≤ HM i ≤ + i=1 Moreover, for every QTM M and every δ, ∆ Q with δ < 1, X ∆, there is a constant c N such that ∈ ≤ M,δ,∆ ∈ ∆ δ since the halting spaces are mutually orthogonal. QCU (ρ) QCM (ρ)+ cM,δ,∆ for every qubit string ρ. Putting classical and quantum part of ϕ together, we get ≤ | i All the proofs are given in Section III, while the ideas of the ϕ˜ := ci ϕ , proofs are outlined in the next subsection. | i ⊗ C| i where i is the halting number of ϕ . Thus, the length of ϕ˜ is exactly . | i | i E. Ideas of Proof n +1 Let sM be a self-delimiting description of the QTM M. The proof of Theorem 1.1 relies on the observation about The idea is to construct a QTM U that, on input sM ϕ˜ , the mutual orthogonality of the halting spaces, as explained proceeds as follows: ⊗ | i in Subsection I-A. Fix some QTM M, and denote the set of 2 n By classical simulation of M, it computes descriptions vectors ψ C ⊗ which cause M to halt at time t by • (n) (n) (n) (n) | i ∈ n of the halting spaces M (1), M (2), M (3),... and (t). If ϕ C2 ⊗ is any halting input for M, then H H H M the corresponding code words c1,c2,c3,... one after the Hwe can decompose| i ∈ ϕ in some sense into a classical and a other, until at step τ, it finds the code word ci that equals quantum part. Namely,| i the information contained in ϕ can | i the code word in the input. be split into a Afterwards, it applies a (quantum) decompression map to • classical part: The vector ϕ is an element of which of approximately reconstruct ϕ from ϕ . • (n) | i the subspaces (t)? Finally, it simulates (quantum)| i for τC|timei steps the time HM • quantum part: Given the halting time τ of ϕ , then where evolution of M on input ϕ and then halts, whatever • (n) | i in the corresponding subspace (τ) is ϕ situated? happens with the simulation.| i HM | i Our goal is to find a QTM U and an encoding ϕ˜ Such a QTM U will have the strong universality property as (n+1) | i ∈ C2 ⊗ of ϕ which is only one qubit longer and which stated in Theorem 1.1. Unfortunately, there are many difficul- makes the (cleverly| i programmed) QTM U output a good ties that have to be overcome by the proof in Section III: approximation of M( ϕ ). First, we extract the quantum part Also classically, QTMs can only be simulated approxi- | i n • out of ϕ . While dim C2 ⊗ = 2n, the halting space mately. Thus, it is for example impossible for U to decide | i STRONGLY UNIVERSAL QUANTUM TURING MACHINES AND INVARIANCE OF KOLMOGOROV COMPLEXITY (AUGUST 11, 2007) 5
by classical simulation whether the QTM M halts on The classical finite binary strings 0, 1 ∗ are identified with { } some input ψ perfectly or only approximately at some the computational basis vectors in 0,1 ∗ , i.e. 0,1 ∗ time t. Thus,| i we have to define certain δ-approximate ℓ2( λ, 0, 1, 00, 01,... ), where λ denotesH{ the} emptyH string.{ } We≃ (n,δ) { } n halting spaces (t) and prove a lot of lemmas with also use the notation n := k and treat it as a HM H≤ k=0 H nasty inequalities. subspace of 0,1 ∗ . H{ } Since our approach includes mixed qubit strings, we have To be as general as possible, weL do not only allow super- • to consider mixed inputs and outputs as well. positions of strings of different lengths, but also mixtures, i.e. The aforementioned prefix code must have the property our qubit strings are arbitrary density operators on 0,1 ∗ . • H{ } that one code word can be constructed after the other It will become clear in the next sections that QTMs naturally (since the sequence of all halting times is not com- produce mixed qubit strings as outputs. Moreover, it will be putable), see Lemma 3.12. a useful feature that the result of applying the partial trace to We show that all these difficulties (and some more) can segments of qubit strings will itself be a qubit string. be overcome, and the idea outlined above can be converted Definition 2.1 (Qubit Strings and their Length): to a formal proof of Theorem 1.1 and the second part of An (indeterminate-length) qubit string σ is a density operator Theorem 1.2 which we give in full detail in Section III. on 0,1 ∗ . Normalized vectors ψ 0,1 ∗ will also be H{ } | i ∈ H{ } For the first part of Theorem 1.2, concerning the complexity called qubit strings, identifying them with the corresponding notion QC, a more general result is needed which is stated density operator ψ ψ . The base length (or just length) of a | +ih | ∗ in Proposition 3.14, since this complexity notion needs an qubit string σ 1 ( 0,1 ) is defined as ∈ T H{ } additional parameter as input. For this proposition, the proof ℓ(σ) := max ℓ(s) s σ s > 0, s 0, 1 ∗ idea outlined above needs to be modified. The idea for the { | h | | i ∈{ } } modified proof of that proposition is to make the QTM U or as ℓ(σ)= if the maximum does not exist. ∞ determine the halting number of the input (and thus the For example, the density operator ϕ ϕ with ϕ as defined halting time) directly by projective measurement in the basis in Equation (3) is a (pure) qubit string| ih of| length| ℓi( ϕ ϕ )= of (approximations of) the halting spaces. We will not prove 5. This corresponds to the fact that this state ϕ | needsih | at Proposition 3.14 in full detail, but only sketch the proof there, least 5 cells on a QTM’s tape to be stored perfectly| i (compare since the technical details are similar to that of the proof of Subsection II-B). An alternative approach would be to consider Theorem 1.1. the expectation value ℓ¯ of the length instead, which has been proposed by Rogers and Vedral [17], see also the discussion II. MATHEMATICAL FRAMEWORK AND FORMALISM in Section IV. Here, we introduce the formalism that is used in Section III In contrast to classical bit strings, there are uncountably to describe qubit strings, quantum Turing machines, and quan- many qubit strings that cannot be perfectly distinguished by tum Kolmogorov complexity. We denote the density operators means of any quantum measurement. A good measure for + on a Hilbert space by ( ) (i.e. the positive trace-class the difference between two qubit strings σ and ρ is the trace H T1 H operators with trace 1). The natural numbers will be denoted distance (cf. [18]) N = 1, 2, 3,..., , and we use the symbols N0 := N 0 {+ } ∪{ } 1 1 and R := x R x 0 as well as δ ′ , which shall be 1 ρ σ Tr := Tr ρ σ = λi , (4) 0 { ∈ | ≥ } t t k − k 2 | − | 2 | | if t = t and 0 otherwise. i ′ X where the λi are the eigenvalues of the trace-class operator A. Indeterminate-Length Qubit Strings ρ σ. Its operational interpretation is that it gives the maximum The quantum analogue of a bit string, a so-called qubit probability− of correctly distinguishing between ρ and σ by string, is a superposition of several classical bit strings. To means of any single quantum measurement. be as general as possible, we would like to allow also superpositions of strings of different lengths like B. Mathematical Description of Quantum Turing Machines 1 ϕ := ( 00 + 11011 ) . (3) Bernstein and Vazirani ([3], Def. 3.2.2) define a quantum √ | i 2 | i | i Turing machine M as a triplet (Σ,Q,δ), where Σ is a finite Such quantum states are called indeterminate-length qubit alphabet with an identified blank symbol #, and Q is a finite strings. They have been studied by Schumacher and West- set of states with an identified initial state q0 and final state Σ Q L,R moreland [15], as well as by Bostr¨om and Felbinger [16] in qf = q0. The function δ : Q Σ C˜ × ×{ } is called the context of lossless quantum data compression. the6 quantum transition function×. The→ symbol C˜ denotes the set 0,1 n Let n := C{ } ⊗ be the Hilbert space of n qubits of complex numbers α C such that there is a deterministic H 0,1 2 ∈ (n N0). We write C{ } for C to indicate that we fix algorithm that computes the real and imaginary parts of α to ∈ two orthonormal computational basis vectors 0 and 1 . The within 2 n in time polynomial in n. | i | i − Hilbert space 0,1 ∗ which contains indeterminate-length One can think of a QTM as consisting of a two-way H{ } qubit strings like ϕ can be formally defined as the direct infinite tape T of cells indexed by Z, a control C, and a | i sum single “read/write” head H that moves along the tape. A ∞ QTM evolves in discrete, integer time steps, where at every 0,1 ∗ := k. H{ } H step, only a finite number of tape cells is non-blank. For Mk=0 STRONGLY UNIVERSAL QUANTUM TURING MACHINES AND INVARIANCE OF KOLMOGOROV COMPLEXITY (AUGUST 11, 2007) 6
every QTM, there is a corresponding Hilbert space QT M = Definition 2.2 (Reading Operation): Q H C T H, where C = C is a finite-dimensional A quantum operation : ( O) ( 0,1 ∗ ) is called a HilbertH ⊗ H space⊗ H spanned byH the (orthonormal) control states reading operation, if forR everyT H finite→T setH of{ classical} strings 2 2 N q Q, while T = ℓ (T ) and H = ℓ (Z) are separable si i=1 0, 1 ∗, it holds that Hilbert∈ spacesH describing the contentsH of the tape and the { } ⊂{ } N position of the head, where . . . # # si # # . . . P αi R -2 -1 0 ℓ(si) ℓ(si) + 1 Z for finitely many (5) i=1 !! T = (xi)i Z Σ xi = # i Z X ∈ ∈ | 6 ∈ N denotes the set of classical tape configurations with finitel y = P α s i| ii many non-blank symbols. i=1 ! For our purpose, it is useful to consider a special class of X where P( ϕ ) := ϕ ϕ denotes the projector onto ϕ . QTMs with the property that their tape T consists of two The condition| i | specifiedih | above does not determine| i different tracks (cf. [3, Def. 3.5.5]), an input track I and an uniquely; there are many different reading operations. For theR output track O. This can be achieved by having an alphabet remainder of this paper, we fix the reading operation which which is a Cartesian product of two alphabets, in our case is specified in the following example. R Σ = 0, 1, # 0, 1, # . Then, the tape Hilbert space T { }×{ } H Example 2.3: Let T denote the classical output track con- can be written as T = I O. figurations as defined in Equation (5), with Σ = 0, 1, # . The transition functionH Hδ ⊗generates H a linear operator U M Then, for every t T , let R(t) be the classical string{ that} on describing the time evolution of the QTM M. If QT M consists of the bits∈ of T from cell number zero to the last δ isH chosen in accordance with certain conditions, then U M non-blank cell, i.e. will be unitary (and thus compatible with quantum theory), + ∗ see Ozawa and Nishimura [5]. We identify σ 1 ( 0,1 ) R : T 0, 1 ∗ ∈ T H{ } →{ } with the initial state of M on input σ, which is according to . . . ? ? s # ? . . . the definition in [3] a state on QT M where σ is written on s. -2 -1 0 ℓ(s) ℓ(s) + 1 7→ the input track over the cell intervalH [0,ℓ(σ) 1], the empty − state # is written on the remaining cells of the input track and For every s 0, 1 ∗, there is a countably-infinite number of ∈{ } on the whole output track, the control is in the initial state q t T such that R(t)= s. Thus, to every t T , we can assign 0 ∈ ∈ and the head is in position 0. By linearity, this e.g. means that a natural number n(t) which is the number of t in some enu- 1 the vector ψ = ( 0 + 11 ) is identified with the vector meration of the set t′ T R(t′)= R(t) ; we only demand | i √2 | i | i { ∈ | } 1 ( 0# + 11 ) on input track cells number 0 and 1. . . . # # s # # . . . √2 that n(t)=1 if t = . | i | i t + -2 -1 0 ℓ(s) ℓ(s) + 1 The global state M (σ) 1 ( QT M ) of M on input σ at 2 2 ∈ T H t t Hence, if (as usual) ℓ ℓ (N) denotes the Hilbert space time t N is given by M t(σ) = (U ) σ (U ) . The state 0 M M∗ of square-summable sequences,≡ then the map U, defined by of the control∈ at time t is thus given by partial trace over all the t t linear extension of other parts of the machine, that is MC(σ) := TrT,H (M (σ)) 2 (similarly for the other parts of the QTM). In accordance with U : O 0,1 ∗ ℓ [3, Def. 3.5.1], we say that the QTM halts at time H → H{ } ⊗ M t N t R(t) n(t) , + ∗ ∈ on input σ 1 ( 0,1 ), if and only if | i 7→ | i ⊗ | i ∈ T H{ } ′ t t is unitary. Then, the quantum operation q MC(σ) q =1 and q MC(σ) q =0 t′ < t, (6) h f | | f i h f | | f i ∀ : ( O) ( 0,1 ∗ ) where qf Q is the final state of the control (specified in R T H → T H{ } ∈ ρ Tr 2 (UρU ∗) the definition of M) signalling the halting of the computation. 7→ ℓ See Subsection I-A for a detailed discussion of these halting is a reading operation. conditions (6). We are now ready to define QTMs as partial maps on the In this paper, when we talk about a QTM, we do not mean qubit strings. the machine model itself, but rather refer to the corresponding Definition 2.4 (Quantum Turing Machine (QTM)): partial function on the qubit strings which is computed by the + ∗ + ∗ A partial map M : 1 ( 0,1 ) 1 ( 0,1 ) will be QTM. Note that this point of view is different from e.g. that { } { } called a QTM, if thereT is aH Bernstein-Vazirani→ T H two-track QTM of Ozawa [9] who describes a QTM as a map from Σ∗ to the M ′ = (Σ,Q,δ) (see [3], Def. 3.5.5) with the following set of probability distributions on Σ∗. properties: We still have to define what is meant by the output of a Σ= 0, 1, # 0, 1, # , QTM M, once it has halted at some time t on some input qubit • { }×{ } t the corresponding time evolution operator UM ′ is unitary, string σ. We could take the state of the output tape MO(σ) • if M ′ halts on input σ at some time t N, then M(σ)= to be the output, but this is not a qubit string, but instead a • t ∈ density operator on the Hilbert space O. Hence, we define M ′O(σ) , where is the reading operation specified H R R a quantum operation which maps the density operators on in Example 2.3 above. Otherwise, M(σ) is undefined. R O to density operators on 0,1 ∗ , i.e. to the qubit strings. A fixed-length QTM is the restriction of a QTM to the domain TheH operation “reads” theH output{ } from the tape. + n N 1 ( n) of length eigenstates. R ∈ 0 T H S STRONGLY UNIVERSAL QUANTUM TURING MACHINES AND INVARIANCE OF KOLMOGOROV COMPLEXITY (AUGUST 11, 2007) 7
The definition of halting, given by Equation (6), is very is with asymptotic probability 1 arbitrarily close to the von important, as explained in Subsection I-A. On the other hand, Neumann entropy rate s of the source. This demonstrates that changing certain details in a QTM’s definition, like the way quantum Kolmogorov complexity is a useful notion, and that to read the output or allowing a QTM’s head to stay at its it is feasible to prove interesting theorems on it. position instead of turning left or right, should not change the While this complexity notion QC(ρ) counts the length results in this paper. of the shortest qubit string that makes a QTM output ρ and halt, there have been different definitions for quantum algorithmic complexity by Vit´anyi [20] and G´acs [21]. Their C. Quantum Kolmogorov Complexity approaches are based on classical descriptions and universal Quantum Kolmogorov complexity has first been defined by density matrices respectively and are not considered in this Berthiaume, van Dam, and Laplante [14]. They define the paper since they do not have the invariance problem outlined complexity QC(ρ) of a qubit string ρ as the length of the in Subsection I-C. shortest qubit string that, given as input into a QTM M, Note also that Definition 2.5 depends on the definition of makes M output ρ and halt. Since there are uncountably many + ∗ the length ℓ(σ) of a qubit string σ 1 ( 0,1 ); there is qubit strings, but a QTM can only apply a countable number a different approach by Rogers and∈ Vedral T H [17]{ that} uses the of transformations (analogously to the circuit model), it is expected (average) length ℓ¯ instead and results in a different necessary to introduce a certain error tolerance δ > 0. notion of quantum Kolmogorov complexity. The results of this This can be done in essentially two ways: First, one can paper are applicable to that definition, too, as long as the notion just fix some tolerance δ. Second, one can demand that the of halting of the corresponding quantum computer is defined QTM outputs the qubit string ρ as accurately as one wants, in a deterministic way as in Equation (6). by supplying the machine with a second parameter as input that represents the desired accuracy. This is analogous to a III. CONSTRUCTION OF A STRONGLY UNIVERSAL QTM classical computer program that computes the number π = A. Halting Subspaces and their Orthogonality 3.14 . . .: A second parameter k N can make the program As already explained in Subsection I-A in the introduction, output π to k digits of accuracy, for∈ example. We consider both restricting to pure input qubit strings of some approaches and follow the lines of [14] except for two simple ψ n fixed length ℓ( ψ )= n, the vectors with| equali ∈ halting H time t modifications: we use the trace distance rather than the fidelity, form a linear subspace| i of . Moreover, inputs with different and we also allow indeterminate-length and mixed input and n halting times are mutuallyH orthogonal, as depicted in Figure 1. output qubit strings. We will now use the formalism for QTMs introduced in Definition 2.5 (Quantum Kolmogorov Complexity): Let M Subsection II-B to give a formal proof of these statements. + ∗ be a QTM and ρ 1 ( 0,1 ) a qubit string. For every { } We use the subscripts C, I, O and H to indicate to what part δ > 0, we define the∈ Tfinite-errorH quantum Kolmogorov com- of the tensor product Hilbert space a vector belongs. plexity QCδ (ρ) as the minimal length of any qubit string M Definition 3.1 (Halting Qubit Strings): + ∗ σ 1 ( 0,1 ) such that the corresponding output M(σ) + { } Let σ ( ∗ ) be a qubit string and M a quantum has∈ trace T H distance from ρ smaller than δ, 1 0,1 Turing∈ machine. T H{ Then,} σ is called t-halting (for M), if M δ QCM (ρ) := min ℓ(σ) ρ M(σ) Tr <δ . halts on input σ at time t N. We define the halting sets and { | k − k } halting subspaces ∈ Similarly, we define the approximation-scheme quantum Kol- mogorov complexity QC (ρ) as the minimal length of any HM (t) := ψ 0,1 ∗ ψ ψ is t-halting for M , M {| i ∈ H{ } | | ih | } + ∗ qubit string σ 1 ( 0,1 ) such that when given M as M (t) := α ψ ψ HM (t), α R , ∈ T H{ } H { | i | | i ∈ ∈ } input together with any integer k, the output M(σ, k) has trace (n) (n) H (t) := HM (t) n, (t) := M (t) n. distance from ρ smaller than 1/k: M ∩ H HM H ∩ H Note that the only difference between H(n)(t) and (n)(t) is 1 M HM QCM (ρ) := min ℓ(σ) ρ M(σ, k) Tr < k N . that the latter set contains non-normalized vectors. It will be k − k k ∀ ∈ (n) shown below that M (t) is indeed a linear subspace. H For the definition of QCM , we have to fix a map to encode Theorem 3.2 (Halting Subspaces): two inputs (a qubit string and an integer) into one qubit string; For every QTM M, n N0 and t N, the sets M (t) and (n) ∈ ∈ H ∗ this is easy, see e.g. [13] for the classical case and [19] for the M (t) are linear subspaces of 0,1 resp. n, and H H{ } H quantum case. Also, using f(k):= 1/k as accuracy required (n) (n) (t) (t′) and (t) (t′) if t = t′. on input k is not important; any other computable and strictly HM ⊥ HM HM ⊥ HM 6 decreasing function f that tends to zero for k such that Proof. Let ϕ , ψ H (t). The property that ϕ is t-halting M ′ 1 → ∞ | i | i ∈ | i t f − is also computable will give the same result up to an is equivalent to the statement that there are states Φ ′ q additive constant. t | i ∈ I O H and coefficients cq C for every t′ t and Note that if M is at least able to move input data to the qH ⊗Q Hsuch⊗ thatH ∈ ≤ δ ∈ output track, then it holds QCM (ρ) ℓ(ρ)+ cM with some t t ≤ VM ϕ I Ψ0 = qf C Φq , (7) constant cM N (and similarly for QCM ). In [19], we have f ′ | i ⊗ | i | i ⊗′ | i ′ ∈ t t t shown that for ergodic quantum information sources, emitted V ϕ I Ψ0 = c q C Φ t′ < t, (8) n M q q 2 ⊗ 1 | i ⊗ | i | i ⊗ | i ∀ states ψ C have a complexity rate n QC• ( ψ ) that q=qf | i ∈ U | i X6 STRONGLY UNIVERSAL QUANTUM TURING MACHINES AND INVARIANCE OF KOLMOGOROV COMPLEXITY (AUGUST 11, 2007) 8
Q where VM is the unitary time evolution operator for the QTM := ϕ n ek ϕ Q + iQ k , where • n n H 2 {| i ∈ H | h | i ∈ ∀ } M as a whole, and Ψ0 = q0 C # O 0 H denotes the ek k=1 denotes the computational basis vectors of n, | i | i ⊗ | i ⊗ | i {| 0 i} ϕ H initial state of the control, output track and head. Note that ϕ := | i for every vector ϕ 0 . • ϕ n does not depend on the input qubit string (in this case | i k| ik | i ∈ H \{ } Ψ0 The set of vectors with rational coordinates, denoted Q, | ).i n ϕ will in the following be used frequently as inputs or outputsH | iAn analogous equation holds for ψ , since it is also t- | i of algorithms. Such vectors can be symbolically added or halting by assumption. Consider a normalized superposition multiplied with rational scalars without any error. Also, given ∗ : α ϕ + β ψ 0,1 a , b Q, it is an easy task to decide unambiguously | i | i ∈ H{ } | i | i ∈ Hn t which vector has larger norm than the other (one can compare VM (α ϕ I + β ψ I) Ψ0 2 2 | i t | i ⊗ | i t the rational numbers a and b , for example). = αV ϕ I Ψ + βV ψ I Ψ M 0 M 0 Now we are readyk | toik statek the | ik main theorem of this | i ⊗ |t i | i ⊗t | i = α q C Φ + β q C Φ˜ | f i ⊗ | qf i | f i ⊗ | qf i subsection: t ˜ t Theorem 3.4 (Computable Approximate Halting Spaces): = qf C α Φq + β Φq . | i ⊗ | f i | f i There is a classical algorithm that, given a classical Thus, the superposition also satisfies condition (7), and, by a description of a QTM M, integers n N0, t N, and a similar calculation, condition (8). It follows that α ϕ + β ψ rational parameter δ > 0, computes a∈ description∈ of some | i | i (n,δ) (n,δ) must also be t-halting. Hence, M (t) is a linear subspace subspace (t) and a rational number ε (t) > 0 H HM ⊂ Hn M of 0,1 ∗ . As the intersection of linear subspaces is again a with the following properties: { } H (n) (n,δ) linear subspace, so must be M (t). Almost-Halting: If ψ H (t), then ψ is (20 δ)-t- H • | i ∈ M | i Let now ϕ HM (t) and ψ HM (t′) such that t on tape cells) to a finite subspace ˜H. Thus, it is possible Proof. Proving this lemma is a routine (but lengthy) exercise. to give a matrix representation of theH time evolution operator The idea is to construct an algorithm that looks for such a VM on C ˜T ˜H, and the expression given above can subspace by brute force, that is, by discretizing the set of all be numericallyH ⊗ H calculated⊗ H just by matrix multiplication and subspaces within some (good enough) accuracy. We omit the subsequent numerical computation of the partial trace. details. 5 Every ϕ that satisfies a(t′) δ ′ ε for every t′ t | ki | − t t|≤ 8 ≤ We proceed by defining approximate halting spaces as will be marked as “approximately halting”. If there is at least the output of a certain algorithm. It will turn out that these one ϕ that is approximately halting, B shall halt and output | ki spaces satisfy all the properties stated in Theorem 3.4. Note 1, otherwise it shall halt and output 0. that the definition depends on the details of the previously To see that this algorithm works as claimed, suppose that defined algorithms in Lemma 3.5 and 3.6 (for example, there Uδ( ϕ ) is not ε-t-halting for M, so for every ψ˜ Uδ( ϕ ) ′ are always different possibilities to compute the necessary | i t | i ∈ | i there is some t′ t such that δ ′ q MC( ψ˜ ψ˜ ) q > discretizations). Thus, we fix a concrete instance of all those ≤ t t − h f | | ih | | f i ε. Also, for every k 1,...,N , there is some vector ψ algorithms for the rest of the paper. ∈{ } 3 | i ∈ Definition 3.7 (Approximate Halting Spaces): Uδ( ϕ ) Sn with ϕk ψ 64 ε, so | i ∩ k | i − | ik ≤ We define3 the δ-approximate halting space (n,δ)(t) ′ M n t (n,δH) ⊂ H ∆k := δt′t qf MC( ϕk ϕk ) qf and the δ-approximate halting accuracy ε (t) Q as the − h | | ih | | i M ∈ ′ t outputs of the following classical algorithm on input n,t δt′t qf MC( ψ ψ ) qf ∈ ≥ − h | | ih | | i N, 0 < δ Q and sM 0, 1 ∗, where sM is a classical ′ ′ ∈ ∈ { } t t 0 0 description of a fixed-length QTM M: qf MC( ψ ψ ) qf qf MC( ϕ ϕ ) qf − h | | ih | | i − h | | kih k| | i (1) Let ε := 18 δ. ′ ′ t t 0 0 qf MC( ϕk ϕk ) qf qf MC( ϕk ϕk ) q f (2) Compute a covering of Sn of open balls of radius δ, that − h | | ih | | i − h | | ih | | i is, a set of vectors ψ ,..., ψ Q (L N) with 0 0 2 1 L n > ε ψ ψ ϕk ϕk Tr 2 1 ϕk δ {| δi | i} ⊂ H ∈ − k | ih | − | ih |k − · − k | ik ψk 1 2 , 1+ 2 for every k 1,...,L such 0 k | ik ∈ L− ∈{ } ε ψ ϕk 2 1 ϕk (1 + ϕk ) that S U ( ψ ). ≥ − k | i − | ik − − k | ik k | ik n i=1 δ i 3 3 23 (3) For every⊂ | i , compute ε ε ϕ ϕ 0 4 ε ε , k 1,...,L B( ψk ,δ,ε,t,sM ) ≥ − 64 − k | ki − | kik − · 64 ≥ 32 and B( ψ S, δ,∈18 { δ,t,s )}, where B is the| algorithmi for | ki M where we have used Lemma A.3 and Lemma A.5. Thus, for testing the ε-t-halting property of balls of Lemma 3.5. If every k it holds the output is 0 for every k, then output ( 0 ,ε) and halt. Otherwise set for N N L and N { K} L ′ 0 0 t ∋ ≤ ∋ ≤ a(t′) δt′t ∆k qf MC( ϕk ϕk ) qf a(t′) ϕ N := ψ B( ψ ,δ,ε,t,s )=1 , − ≥ − h | | ih | | i− {| ii}i=1 {| ki | | ki M } 23 3 5 K > ε ε = ε , ϕ˜i i=1 := ψk B( ψk , δ, 18 δ,t,sM )=0 . 32 − 32 8 {| i} {| i | | i } If N = 0, i.e. if the set ϕ N is empty, output which makes the algorithm halt and output 0. {| ii}i=1 ε ( 0 ,ε) and halt. On the other hand, suppose that Uδ( ϕ ) is 4 -t-halting for { } n | i ε (4) Set d := 2 . M, i.e. there is some ψ Uδ( ϕ ) Sn which is 4 -t- ˜ 7 ˜ 3 | i ∈ | i ∩ (5) Let ∆ := 2δ, ∆ := 4 δ and δ := 2 δ. halting for M. By construction, there is some k such that Use the algorithm I of Lemma 3.6 to search 3 . A similar calculation as above yields ϕk ψ 64 ε for an interpolating subspace, i.e., compute k | i − | ik′ ≤ 17 ′ t ˜ ˜ δt t qf MC( ϕk ϕk ) qf 32 ε for every t′ t, so I(K,N, ϕ˜1 ,..., ϕ˜K , ϕ1 ,..., ϕN , d, ∆, δ, ∆, δ). If − h | |17 ih 3| | i 5≤ ≤ | i | i | i | i a(t′) δ ′ ε + ε = ε, and the algorithm outputs the output of I is (1, U˜), output U,ε˜ and halt. − t t ≤ 32 32 8 1. (6) Set . If , then go back to step (5). d := d 1 d 1 (7) Set ε := ε −and go back≥ to step (3). Lemma 3.6 (Algorithm I for Interpolating Subspace): 2 Moreover, let H(n,δ)(t) := (n,δ)(t) S . There exists a (classical) algorithm I which, on input M,N M HM ∩ n Q + ∈ The following theorem proves that this definition makes N, ϕ˜1 ,..., ϕ˜M , ϕ1 ,..., ϕN n , d N, Q ∆ > δ and| Qi + |∆˜ >i δ˜|, alwaysi halts| i and ∈ H returns∈ the description∋ sense: ∋ Theorem 3.8: The algorithm in Definition 3.7 always ter- of a pair (i, U˜) with i 0, 1 and U˜ n a linear subspace, ∈{ } ⊂ H minates on any input; thus, the approximate halting spaces under the following constraints: (n,δ) M (t) are well-defined. If the output is (1, U˜), then U˜ n must be a subspace H + • Proof. Define the function ε : S R by ε ( ψ ) := of dimension dim U˜ = d such⊂ that H dist(U,˜ ϕ ) < ∆ min n 0 min k inf ε> 0 ψ is ε-t-halting for M →. Lemma A.3 and| i A.5 for every k and dist(U,˜ ϕ˜ ) > δ˜ for every l|. i l yield{ | | i } If there exists a subspace| i U of dimension • ⊂ Hn dim U = d such that dist(U, ϕ ) δ for every k and εmin( ψ1 ) εmin( ψ2 ) ψ1 ψ2 , (9) k | i − | i ≤ k | i − | ik dist(U, ϕ˜ ) ∆˜ for every l,| theni the≤ output must be of l 2 ˜ 2 | i ≥˜ U will then be an approximation of U . the form (1, U). 3 (n,δ) From a formal point of view, the notation should rather read HsM (t) ˜ n,δ The description of the subspace U is a list of linearly inde- instead of H( )(t), since this space depends also on the choice of the d Q M pendent vectors u˜ U˜. classical description sM of M. {| ii}i=1 ⊂ Hn ∩ STRONGLY UNIVERSAL QUANTUM TURING MACHINES AND INVARIANCE OF KOLMOGOROV COMPLEXITY (AUGUST 11, 2007) 10 (n) (n,δ) so εmin is continuous. For the special case HM (t) = , it (1, M (t)). By definition of I, it holds must thus hold that ε (S ) := min ε ( ψ ) >∅ 0. H (n,δ) min n ψ Sn min dist( M (t), ψj ) < ∆, and by elementary estimations it | i∈ | i H | i(n,δ) If the algorithm has run long enough such that ε<εmin(Sn), δ follows that dist( M (t) Sn, ψj ) < 2 + 2∆, so there is it must then be true that B( ψk ,δ,ε,t,sM )=0 for every (δ) (Hn,δ) ∩ | i (δ) δ | i some ψ HM (t) such that ψ ψj < 2 + 2∆. k 1,...,L , since all the balls Uδ( ψk ) are not ε-t-halting. | i ∈ k | i − | ik ∈{ } | i Since ψ ψj δ and ∆=2δ, the approximation This makes the algorithm halt in step (3). propertyk | follows.i − | ik ≤ Now consider the case H(n)(t) = . The continuous M 6 ∅ Notice that under the assumptions given in the statement function εmin attains a minimum on every compact set of the similarity property, it follows from the almost-halting U¯ ( ψ ) S , so let ε := min ¯ ε ( ψ ) (n,δ) 1 (n,∆) δ k n k ψ Uδ( ψk ) Sn min property that if ψ H (t), then ψ must be ε (t)- (1 | ki ∩N). If ε = 0 for every| ki∈, then| fori ∩ every k |andi | i ∈ M | i 4 M k t-halting for M. Consider the computation of (n,∆)(t) by ε >≤ 0,≤ there is some vector ψ U ( ψ ) S which M δ k n the algorithm in Definition 3.7. By construction,H it always is ε-t-halting for M, so B( ψ| ,δ,ε,t,si ∈ | )=1i ∩ for every k M holds that the parameter ε during the computation satisfies ε > 0, and so K = 0 in step| (3).i Thus, the algorithm I will ε ε(n,∆)(t), so ψ is always ε -t-halting for M, and if by construction find the interpolating subspace U˜ = and M 4 n ≥ , it| followsi that . cause halting in step (5). H ψ Uδ( ψj ) B( ψj ,δ,ε,t,sM ) = 1 The| i ∈rest follows| i in complete analogy| toi the proof of the Otherwise, let ε := min ε k 1,...,N ,ε > 0 . 0 k k approximation property. Suppose that the algorithm{ has| run∈ { long enough} such} For the almost-orthogonality property, suppose that ε < ε0. By construction of the algorithm B, if v H(n,δ)(t ) and w H(n,δ)(t) are two arbitrary B( ψ ,δ,ε,t,s ) = 1, it follows that U ( ψ ) is ε-t- M ′ M k M δ k qubit| i ∈ strings of length | withi ∈ different approximate halting halting| i for M, but then, ε ε < ε , so |ε i= 0, so n k 0 k times . There is some such that there is some ψ U¯ ( ψ ≤) S which is 0-t-halting t n n n of Theorem 3.4, it follows that ϕi ϕj 4√5δ < 2− = Let V be the unitary 2 2 -matrix that has the vectors 1 |h | i| ≤ n 2n × (2n+1) 1 for every i = j. Lemma A.1 yields dim U 2 +1 ui i=1 as its column vectors. The algorithm continues by − 6 n ≥ {| i} ˜ for U := span ϕ 2 +1 , but dim =2n, which is computing a rational approximation V of V such that the i i=1 n n δ {| i} ⊂ H H entries satisfy V˜ V < n , and thus, in a contradiction. | ij − ij | 2n+1(10√2n)2 δ operator norm, it holds V˜ V < n . Bernstein k − k 2(10√2n)2 and Vazirani [3, Sec. 6] have shown that there are QTMs C. Compression, Decompression, and Coding that can carry out an ε-approximation of a desired unitary In this subsection, we define some compression and coding transformation V on their tapes if given a matrix V˜ as input algorithms that will be used in the construction of the strongly that is within distance ε of the d d-matrix V . This 2(10√d)d × universal QTM. is exactly the case here5, with d = 2n and ε = δ, so let the Definition 3.10 (Standard (De-)Compression): D apply V within δ on its tape to create the state ϕ | i ∈ Hn Let U n be a linear subspace with dim U = N. Let with ϕ V ψ˜ = ϕ V A ψ < δ. Note that the ⊂ H k | i− | ik k | i− ◦ | ik PU ( n) be the orthogonal projector onto U, and let map V A is a standard decompression map (as defined in ∈2n B H ◦ ei i=1 be the computational basis of n. The result of Definition 3.10), since for every i 1,..., dim U it holds {|applyingi} the Gram-Schmidt orthonormalizationH procedure to that ∈ { } 2n 2n the vectors u˜i i=1 = PU ei i=1 (dropping every null vector) is called{| thei} standard{ basis| i} u ,..., u of U. Let V A U ui = V A fi = V ei = ui , {| 1i | N i} ◦ ◦C | i ◦ | i | i | i fi be the i-th computational basis vector of log N . The where the vectors fi are the computational basis vectors of standard| i compression is thenH⌈ defined⌉ by | i U : U log N log dim U . C → H⌈ ⌉ ⌈ ⌉ linear extension of U ( ui ) := fi for 1 i N, that is, U H C | i | i ≤ ≤ C The next lemma will be useful for coding the “classical part” isometrically embeds U into log N . A linear isometric map H⌈ ⌉ of a halting qubit string. The “which subspace” information U : log N n will be called a standard decompression D H⌈ ⌉ → H will be coded into a classical string c 0, 1 ∗ whose length if it holds that i ∈{ } 1 ℓi N0 depends on the dimension of the corresponding U U = U . ∈ (n,δ) D ◦C halting space M (ti). The dimensions of the halting spaces It is clear that there exists a classical algorithm that, given a de- (n,δ) H (n,δ) dim (t1), dim (t2),... can be computed one dim U Q HM HM scription of U (e.g. a list of basis vectors ui i=1 n ), after the other, but the complete list of the code word lengths can effectively compute (classically) an approximate{| i} desc⊂ Hrip- ℓi is not computable due to the undecidability of the halting tion of the standard basis of U. Moreover, a quantum Turing problem. Since most well-known prefix codes (like Huffman machine can effectively apply a standard decompression map code, see [22]) start by initially sorting the code word lengths to its input: in decreasing order, and thus require complete knowledge of Lemma 3.11 (Q-Standard Decompression Algorithm): the whole list of code word lengths in advance, they are D 4 There is a QTM which, given a description of a subspace not suitable for our purpose. We thus give an easy algorithm U , the integer n N, some δ Q+, and a quantum ⊂ Hn ∈ ∈ that constructs the code words one after the other, such that state ψ log dim U , outputs some state ϕ n with the | i ∈ H⌈ ⌉ | i ∈ H code word ci depends only on the previously given lengths property that ϕ U ψ <δ, where U is some standard k | i−D | ik D ℓ1,ℓ2,...,ℓi. We call this “blind prefix coding”, because code decompression map. words are assigned sequentially without looking at what is Proof. Consider the map A : log dim U n, given coming next. (n log dim U ) H⌈ ⌉ → H by A v := 0 ⊗ −⌈ ⌉ v . The map A prepends | i | i ⊗ | i Lemma 3.12 (Blind Prefix Coding): zeroes to a vector; it maps the computational basis vectors of N Let ℓi i=1 N0 be a sequence of natural numbers (code log dim U to the lexicographically first computational basis { } ⊂ N H⌈ ⌉ ℓi vectors of n. The QTM D starts by applying this map A to word lengths) that satisfies the Kraft inequality 2− 1. H ≤ the input state ψ by prepending zeroes on its tape, creating i=1 | i (n log dim U ) Then the following (“blind prefix coding”) algorithmX produces a state ψ˜ := 0 ⊗ −⌈ ⌉ ψ . n N | i | i ⊗ | i ∈ H a list of code words c 0, 1 ∗ with ℓ(c ) = ℓ , such Afterwards, it applies (classically) the Gram-Schmidt { i}i=1 ⊂ { } i i orthonormalization procedure to the list of vectors that the i-th code word only depends on ℓi and the previously Q chosen codewords c1,...,ci 1: u˜1 ,..., u˜dim U , e1 ,..., e2n n , where the − {| i | dim Ui | i | i} ⊂ H Start with c := 0ℓ1 , i.e. c is the string consisting of ℓ vectors u˜i i=1 are the basis vectors of U given in the • 1 1 1 {| i} 2n zeroes; input, and the vectors ei i=1 are the computational basis {| i} for i = 2,...,N recursively, let c be the first string in vectors of n. Since every vector has rational entries (i.e. • i H Q is an element of ), the Gram-Schmidt procedure can be lexicographical order of length ℓ(ci)= ℓi that is no prefix n n applied exactly, resultingH in a list u 2 of basis vectors or extension of any of the previously assigned code words {| ii}i=1 of which have entries that are square roots of rational c1,...,ci 1. Hn − numbers. Note that by construction, the vectors u dim U Proof. We omit the lengthy, but simple proof; it is based {| ii}i=1 are the standard basis vectors of U that have been defined in on identifying the binary code words with subintervals of Definition 3.10. 5 Note that we consider Hn as a subspace of an n-cell tape segment Hilbert 4 Q {0,1,#} ⊗n (a list of linearly independent vectors {|u˜1i,..., |u˜ i} ⊂ U ∩ Hn) space C , and we demand V to leave blanks |#i invariant. dim U ` ´ STRONGLY UNIVERSAL QUANTUM TURING MACHINES AND INVARIANCE OF KOLMOGOROV COMPLEXITY (AUGUST 11, 2007) 12 N [0, 1) as explained in [13]. We also remark that the content (M,n) to the sequence ℓi which has been constructed in of this lemma is given in [22, Thm. 5.2.1] without proof as i=1 Lemma 3.12. n o an example for a prefix code. (n,ε0) In the following, we use the space M (t) as some kind of “reference space” i.e. we constructH our QTM U such that (n,ε ) D. Proof of the Strong Universality Property it expects the standard compression of states ψ 0 (t) | i ∈ HM To simplify the proof of Main Theorem 1.1, we show now as part of the input. If the desired accuracy parameter δ is that it is sufficient to consider fixed-length QTMs only: smaller than ε0, then some “fine-tuning” must take place, unitarily mapping the state ψ (n,ε0)(t) into halting Lemma 3.13 (Fixed-Length QTMs are Sufficient): | i ∈ HM For every QTM M, there is a fixed-length QTM M˜ such that spaces of smaller accuracy parameter. In the next paragraph, + ∗ these unitary transformations are constructed. for every ρ 1 ( 0,1 ) there is a fixed-length qubit string { } 1 (n,εk−1) ∈ T+ H ˜ Recursively, for k N, define εk := ε (t). ρ˜ n N 1 ( n) such that M(ρ) = M(˜ρ) and ℓ(˜ρ) 80 M ∈ ∈ 0 T H ≤ (n,δ) ∈ ℓ(ρ)+1. Since εM (t) 18δ by construction of the algorithm in S n+1 ≤ 18 k k Proof. Since dim n = 2 1, there is an isometric Definition 3.7, we have ε ε →∞ 0. It follows H≤ − k ≤ 80 · 0 −→ embedding of n into n+1. One example is the map Vn, from the approximation property of Theorem 3.4 together with H≤ H n+1 (n,ε ) (n) which is defined as Vn ei := fi for i 1,..., 2 1 , Lemma A.4 that k . The similarity | i | i ∈{ − } dim M (t) dim M (t) where ei and fi denote the computational basis vectors H ≥ H (n,εk−1) | i | i property and Lemma A.4 tell us that dim M (t) (in lexicographical order) of n and n+1 respectively. As (n,εk) H ≥ H≤ H dim M (t) for every k N, and there exist isometries n+1 (n+1) and n (n+1), we can extend H (n,εk ) (n,εk−∈1) H ⊂ H≤ H≤ ⊂ H≤ Uk : M (t) M (t) that, for k large enough, Vn to a unitary transformation Un on (n+1), mapping H → H ≤ satisfy computational basis vectors to computationalH basis vectors. 2n k The fixed-length QTM M˜ works as follows, given some 2 1 8 11 5 18 fixed-length qubit string + on its input tape: Uk < εk 1 constn . (11) ρ˜ 1 ( n+1) k − k 3 2 − 2 ≤ · 80 first, it determines n +1=∈ℓ(˜ρ T) byH detecting the first blank r (n,εk ) symbol #. Afterwards, it computes a description of the unitary Let now d := limk dim (t) and c := →∞ M (n,εk) H transformation Un∗ and applies it to the qubit string ρ˜ by min k N dim (t)= d . For any choice of the ∈ | HM permuting the computational basis vectors in the (n+1)-block transformations U (they are not unique), let (n+1) n k o of cells corresponding to the Hilbert space C 0,1,# ⊗ . { } (n,εc) (n,ε ) U U ...U (t) if k If σ = λ ψ ψ is a mixed fixed-length qubit string U : (n,εk)(t) (n,εk−1)(t) such that k k| kih k| k HM → HM which is τ-halting for M, every convex component ψk P | i N N ε must also be τ-halting for M, and it makes sense to define ˜ ˜(n,ε0) Uk∗ Vk∗ ↾ M (t) < . (M,n) (M,n) (M,n) (M,n) − H 2 σ := k λk ψk ψk , where every ψk k=1 k=1 (M,n) | ih | | i Y Y (and thus σ ) starts with the same classical code word N (M,n) P (M,n) + Setting V := k=1 Vk∗ will work as desired, since ci , and still σ 1 ( n+1). ∈ T H N The strongly universal QTM U expects input of the form ∞ ˜ Q 1 Uk∗ U Uk − ≤ k − k (M,n) (12) kY=1 k=XN+1 sM σ ,δ =: (σM ,δ) , k ⊗ ∞ 18 2 ε constn < where s 0, 1 is a self-delimiting description of the ≤ · 80 2 M ∗ k=N+1 QTM M. We∈ { will} now give a description of how U works; X due to Equation (11) and the proof of Lemma A.2. meanwhile, we will always assume that the input is of the Use V to carry out a δ -approximation of a unitary expected form (12) and also that the input σ is a pure qubit • 3 extension U˜ of U on the state ϕ˜ on the tape (the string ψ ψ (we discuss the case of mixed input qubit strings reason why this is possible is explained| i in the proof σ afterwards):| ih | of Lemma 3.11). This results in a vector ϕ with the Read the parameter δ and the description sM . ˜ δ | i • property that ϕ U ϕ˜ < 3 . Look for the first blank symbol # on the tape to deter- k | i− | ik • Simulate M on input ϕ ϕ for τ time steps within an (M,n) • δ | ih | mine the length ℓ(σ )= n +1. O accuracy of 3 , that is, compute an output track state ρ + τ δ ∈ Compute the halting time τ. This is achieved as follows: ( O) with ρO MO( ϕ ϕ ) < , move this • T1 H k − | ih | kTr 3 (1) Set t := 1 and i := 0. state to the own output track and halt. (It has been shown (n,ε0) (2) Compute a description of M (t). If by Bernstein and Vazirani in [3] that there are QTMs that (n,ε0) H can do a simulation in this way.) dim M (t)=0, then go to step (5). H (M,n) (M,n) (3) Set and set Let σM := sM σ . Using the contractivity of the trace i := i + 1 ℓi := n + 1 ⊗ (n,ε0) − distance with respect to quantum operations and Lemma A.3, log dim (t) . From the previously com- HM we get lputed code word lengthsm ℓ(M,n) (1 j i), j ≤ ≤ U (σ ,δ) M( ψ ψ ) = compute the corresponding blind prefix code word k M − | ih | kTr (M,n) (M,n) τ = (ρO) (MO( ψ ψ )) ci . Bit by bit, compare the code word ci Tr (M,n) kR −τ R | ih | k with the prefix of σ . As soon as any difference ρO MO( ϕ ϕ ) Tr is detected, go to step (5). ≤ k −τ | ih | k τ + MO( ϕ ϕ ) MO( ψ ψ ) (4) The halting time is . Exit. k | ih | − | ih | kTr τ := t δ (5) Set t := t +1 and go back to step (2). < + ϕ ϕ ψ ψ 3 k| ih | − | ih |kTr Let ψ˜ be the rest of the input, i.e. • δ (M,n)| i (M,n) (M,n) ˜ ˜ + ϕ ψ σ =: ci ci ψ ψ (thus ≤ 3 k | i − | ik iθ | 1 ih | ⊗ | ih | ˜ (n,ε ) with some irrelevant phase δ ψ = e 0 (τ)U − ψ | i C M | i + ϕ U˜ ϕ˜ + U˜ ϕ˜ ψ θ R). ApplyH the quantum standard decompression ≤ 3 k | i− | ik k | i − | ik algorithm∈ D given in Lemma 3.11, i.e. compute 2 < δ + ϕ˜ U˜ ∗ ψ < δ . ϕ˜ := D (n,ε0)(τ), n, δ , ψ˜ . Then, 3 | i− | i M 3 | i H | i This proves the claim for pure inputs σ = ψ ψ . If | ih | σ = λk ψk ψk is a mixed qubit string as explained 1 δ k (n,ε ) ˜ − | ih | ϕ˜ 0 (τ) ψ = ϕ˜ U ψ < . right before Equation (12), the result just proved holds for | i − DHM | i | i− | i 3 P every convex component of σ by the linearity of M, i.e. Compute an approximation V : n n of a ρ M( ψ ψ ) <δ, and the assertion of the theorem • H → H k k k Tr unitary extension of U with U V ↾ ˜(n,ε0)(τ) < kfollows− from| theih joint| k convexity of the trace distance and the − HM δ/3 observation that U takes the same number of time steps for n =: ε, where U is some “fine-tuning map” 2(10√2n)2 every convex component ψ ψ . as constructed above. This can be achieved as follows: | kih k| This proof relies on the existence of a universal QTM in – Choose N N large enough such that U k the sense of Bernstein and Vazirani as given in Equation (1). ∈ 18 2 ε ∞ , where R k=N+1 constn 80 < 2 constn Nevertheless, the proof does not imply that every QTM that is the constant defined· in Equation (11). ∈ P satisfies (1) is automatically strongly universal in the sense of – For every k 1,...,N , find matrices Vk : n Theorem 1.1; for example, we can construct a QTM that that approximate∈{ the} forementioned6 isometriesH → U Hn always halts after T simulated steps of computation on input (sM , T, δ, ψ ) and that does not halt at all if the input is not 6 | i The isometries Uk are not unique, so they can be chosen arbitrarily, except of this form. So formally, for the requirement that Equation (11) is satisfied, and that every Uk depends (n,ε − ) only on H(n,εk )(t) and H k 1 (t) and not on other parameters. QTM universal by (1) ) U QTM strongly universal . M M {U } { } STRONGLY UNIVERSAL QUANTUM TURING MACHINES AND INVARIANCE OF KOLMOGOROV COMPLEXITY (AUGUST 11, 2007) 14 Proposition 3.14 (Parameter Strongly Universal QTM): into detail; we will just claim that such a definition can be There is a fixed-length quantum Turing machine U with found in a way such that these δ-approximations share enough the property of Theorem 1.1 that additionally satisfies the properties with their counterparts from Definition 3.7 to make following: For every QTM M and every qubit string σ the algorithm given below work. + ∗ + ∗∈ 1 0,1 , there is a qubit string σM 1 0,1 We are now going to describe how a machine U with the T H{ } ∈ T H{ } such that properties given in the assertion of the proposition works. (M,n) 1 It expects input of the form k,f sM σ , where U (σM , k) M (σ, 2k) < for every k N ⊗ ⊗ k − kTr 2k ∈ f 0, 1 is a single bit, sM 0, 1 ∗ is a self-delimiting ∈ { } ∈(M,n { ) } + description of the QTM , ∗ is a if M(σ, 2k) is defined for every k N, where the length of M σ 1 ( 0,1 ) qubit string, and N an arbitrary integer.∈ T H For{ the} same σ is bounded by ℓ(σ ) ℓ(σ)+∈ c , and c N is a k M M M M reasons as in the proof∈ of Theorem 1.1, we may without loss constant depending only on≤M. ∈ of generality assume that the input is a pure qubit string, so One might first suspect that this proposition is an easy (M,n) (M,n) (M,n) . Moreover, due to Lemma 3.13, corollary of Theorem 1.1, but this is not true. The problem σ = ψ ψ we may also| assumeih that M| is a fixed-length QTM, and so is that the computation of M(σ, k) may take a different σ(M,n) +( ) is a fixed-length qubit string. number of time steps τ for different k (typically, τ 1 n ∈ T H U for k ). Just using the result of Theorem 1.1 would→ give ∞ These are the steps that performs: → ∞ a corresponding qubit string σM that depends on k, but here (1) Read the first bit f of the input. If it is a 0, then proceed we demand that the qubit string σM is the same for every with the rest of the input the same way as the QTM that k, which is important for the proof of Theorem 1.2 to fit the is given in Theorem 1.1. If it is a 1, then proceed with definition of QC. the next step. This ensures that the resulting QTM U will Thus, we have to give a new proof that is different from still satisfy the statement of Theorem 1.1. the proof of Theorem 1.1. Nevertheless, the new proof relies (2) Read sM , read k, and look for the first blank symbol # essentially on the same ideas and techniques; for this reason, to determine the length n := ℓ(σ(M,n)). + we will only sketch the proof and omit most of the details. (3) Set j := 1 and δ0 Q (depending on n) small enough. The proof sketch is based on the idea that a QTM which (4) Set t := 1. ∈ is universal in the sense of Bernstein and Vazirani (i.e. as in (n,δ0) U (5) Compute M (τ1,...,τj 1,t). Find a -exact projec- Equation (1)) has a dense set of unitaries that it can apply (n) H − tor P (τ1,...,τj 1,t) with the following properties: U M − exactly. We can call such unitaries on n for n N -exact (n) (n) H ∈ P (τ1,...,τj 1,t′) P (τ1,...,τj 1,t)=0 for unitaries. • M − · M − This follows from the result by Bernstein and Vazirani that every 1 t′ N 1 Thus, it seems that at least from the point of view of quantum Proof. We first show by induction that k=1 Uk δ − ≤ Kolmogorov complexity QC , it is sufficient to consider N U 1 . This is trivially true for N = 1; suppose it k=1 k Q machines with a completely classical control. Such machines is truek for N− factors,k then do not have the problem of “approximate halting” described P in Subsection I-A. N+1 N+1 N N Uk 1 Uk Uk + Uk 1 There may be interesting applications of extending algorith- − ≤ − − mic information theory to the quantum case. One exciting per- kY=1 kY=1 kY=1 kY=1 N N spective is that in a quantum theory of algorithmic complexity, (UN+1 1) U k + Uk 1 both the inherent notions of “randomness” of quantum theory ≤ − k − k kY=1 kX=1 and “algorithmic randomness” originating from undecidability N +1 results will occur (and maybe be related) in a single theory. U 1 . ≤ k k − k One possible application of quantum Kolmogorov complexity k=1 might be to analyze a fully quantum version of the thought X By assumption, the sequence a := n U 1 is a experiment of Maxwell’s demon in statistical mechanics, since n k=1 k k − k Cauchy sequence; hence, for every ε> 0 there is an Nε N its classical counterpart has already proved useful for the P N ∈ such that for every L,N Nε it holds that k=L+1 Uk corresponding classical analysis (cf. [13]). ≥ n k − 1 <ε. Consider now the sequence Vn := Uk. If N k Pk=1 ≥ L Nε, then APPENDIX ≥ Q N L L Lemma A.1 (Inner Product and Dimension Bound): VN VL = Uk Uk Uk k − k · − Let be a Hilbert space, and let ψ1 ,..., ψN with k=YL+1 kY=1 kY=1 H | i | i ∈ H N L ψi = 1 for every i 1,...,N , where 2 N N. k | ik ∈ { } ≤ ∈ 1 Suppose that Uk Uk ≤ − · k=YL+1 kY=1 1 N ψi ψj < for every i = j . h | i N 1 6 Uk 1 < ε , − ≤ k − k Then, dim N. k=L+1 H≥ X Proof. We prove the statement by induction in N N. For so (Vn)n N is also a Cauchy sequence and converges in ∈ ∈ N = 2, the statement of the theorem is trivial. Suppose the operator norm to some linear operator U on V1. It is easily claim holds for some N 2, then consider N +1 normalized checked that U must be isometric. vectors ψ ,..., ψ ≥ , where is an arbitrary Hilbert 1 N+1 Lemma A.3 (Norm Inequalities): Let be a finite- space. Suppose| i that| ψiψ ∈ H< 1 forH every i = j. Let P := i j N dimensional Hilbert space, and let H with 1 ψ ψ , then|h |P ψi| =0 for every i 6 1,...,N , ψ , ϕ −| N+1ih N+1| | ii 6 ∈{ } ψ = ϕ =1. Then, | i | i ∈ H and let k | ik k | ik ϕi′ ψ ψ ϕ ϕ ψ ϕ . ϕi′ := P ψi , ϕi := | i . k | ih | − | ih |kTr ≤ k | i − | ik | i | i | i ϕi′ k | ik Moreover, if ρ, σ +( ) are density operators, then ∈ T1 H The ϕi are normalized vectors in the Hilbert subspace ˜ := | i 2 2 H ρ σ ρ σ Tr . ran(P ) of . Since ϕi′ = ψi ψi ψi ψN+1 > 1 k − k≤k − k 1 H k | ik h | i−|h | i| − N 2 , it follows that the vectors ϕi have small inner product: Proof. Let ∆ := ψ ψ ϕ ϕ . Using [18, 9.99], Let i = j, then | i | ih | − | ih | 6 ∆ 2 = 1 ψ ϕ 2 = 1 ψ ϕ 1+ ψ ϕ 1 k kTr − |h | i| − |h | i| |h | i| ϕ ϕ = ϕ ϕ i j i′ j′ 2 |h | i| ϕi′ ϕj′ |h | i| ≤ k | ik · k | ik 2 2 ψ ϕ 2 2Re ψ ϕ ψi ψj ψN+1 ψj ψi ψN+1 ≤ − |h | i| ≤ − h | i| {z } < |h | i − h | ih | i| = ψ ϕ ψ ϕ = ψ ϕ 2 . 1 1 h − | − i k | i − | ik 1 N 2 1 N 2 − − Let now ∆˜ := ρ σ, then ∆˜ is Hermitian. We may assume 1 q 1 q1 1 − < + = . that one of its eigenvalues which has largest absolut value is 1 1 N N 2 N 1 − N 2 − positive (otherwise interchange ρ and σ), thus Thus, dim ˜ N, and so dim N +1. ∆˜ = max v ∆˜ v = max Tr(P ∆)˜ H≥ H≥ k k v =1h | | i k| ik P proj., TrP =1 Lemma A.2 (Composition of Unitary Operations): max Tr(P ∆)˜ = ∆˜ Tr Let be a finite-dimensional Hilbert space, let (Vi)i N be a ≤ P proj. k k sequenceH of linear subspaces of (which have all the∈ same according to [18, 9.22]. dimension), and let U : V V H be a sequence of unitary i i → i+1 operators on such that ∞ U 1 exists. Then, the Lemma A.4 (Dimension Bound for Similar Subspaces): H k=1 k k − k product ∞ Uk = . . . U3 U2 U1 converges in operator- Let be a finite-dimensional Hilbert space, and let V, W k=1 · P· · H ⊂ H norm to an isometry U : V1 . be subspaces such that for every v V with v = 1 Q → H | i ∈ k | ik STRONGLY UNIVERSAL QUANTUM TURING MACHINES AND INVARIANCE OF KOLMOGOROV COMPLEXITY (AUGUST 11, 2007) 17 there is a vector w W with w = 1 which sat- Now define the linear operator U : V W via linear exten- isfies v w | i ∈ε, where 0 k< | εik 1 is sion of U v := e for 1 i d. This→ map is an isometry, k | i − | ik ≤ ≤ 4(dim V 1)2 | ii | ii ≤ ≤ fixed. Then, dim W dim V . Moreover, if additionally− since it maps an orthonormal basis onto an orthonormal basis 2 2 dim V ≥ d 1 5 − of same dimension. By substituting v = α v and ε 36 2 holds, then there exists an isometry k=1 k k ≤ dim V | i | i U : V W such that U 1 < 8 √ε 5 . using ε< 4√ε and the geometric series, it easily follows that 3 2 8 5 d P Proof.→Let v ,..., vk −bek an orthonormal basis of V . By U v v √ε if v =1. 1 d k | i − | ik ≤ 3 2 k | ik assumption,{| therei are| normalizedi} vectors w ,..., w {| 1i | di} ⊂ Lemma A.5 (Stability of the Control State): W with vi wi ε for every i. From the definition If ψ , ϕ , v and ψ = ϕ = 1 and v = 0, of the tracek | distancei − | ik for ≤ pure states (see [18, (9.99)] together | i | i | i ∈ Hn k | ik k | ik | i 6 then it holds for every QTM M and every t N0 with Lemma A.3, it follows for every i = j ∈ 6 t t qf MC( ψ ψ ) qf qf MC( ϕ ϕ ) qf 1 w w 2 = w w w w h | | ih | | i − h | | ih | | i − |h i| j i| k | iih i| − | j ih j |kTr ψ ψ ϕ ϕ , ≤ | ih | − | ih | Tr q vi vi vj vj Tr t t 0 0 ≥ k | ih | − | ih |k qf MC ( v v ) qf qf MC( v v ) qf vi vi wi wi Tr h | | ih | | 2 i − h | | ih | | i −k | ih | − | ih |k 1 v . vj vj wj wj Tr ≤ − k | ik −k | ih | − | ih |k Proof. Using the Cauchy-Schwarz inequality, Lemma A.3 and 1 v w v w ≥ − k | ii − | iik − k | ji − | j ik the contractivity of quantum operations with respect to the 1 2ε . ≥ − trace distance (cf. [18, (9.35)]), we get the chain of inequalities 1 Thus, wi wj < 2√ε , and it follows from t t d 1 ∆ := q MC( ψ ψ ) q q MC( ϕ ϕ ) q Lemma|h A.1| thati| dim W d≤. Now− apply the Gram-Schmidt t f f f f h |t | ih | | it − h | | ih | | i ≥ d MC ψ ψ MC ϕ ϕ orthonormalization procedure to the vectors wi i=1: {| i} ≤ t | ih | − t | ih | MC ψ ψ MC ϕ ϕ k 1 Tr − e˜k ≤ | ih | − | ih | e˜k := wk wk ei ei , ek := | i . ψ ψ ϕ ϕ . | i | i− h | i| i | i e˜ ≤ | ih | − | ih | Tr i=1 k | kik X The second inequality can be proved by an analogous calcu- Use e˜k 1 = e˜k wk e˜k wk and lation. calculatek | ik − k | ik − k | ik ≤ k | i − | ik k 1 − wk e˜i e˜i e˜ w = h | i| i ACKNOWLEDGMENT k | ki − | kik e˜ 2 i=1 i X k | ik Sincere thanks go to Wim van Dam, Caroline Rogers, k 1 − wk e˜i wi + wk wi Torsten Franz, and David Gross for helpful discussions, and to |h | − i| |h | i| ≤ e˜ an anonymous referee for very useful comments on a previous i=1 k | iik kX1 draft. Also, the author would like to thank his collegues Ruedi − e˜ w +2√ε k | ii − | iik . Seiler, Arleta Szkoła, Rainer Siegmund-Schultze, Tyll Kr¨uger, ≤ 1 e˜ w i=1 i i and Fabio Benatti for constant support and encouragement. X − k | i − | ik Let ∆k := e˜k wk for every 1 k d. We will k | i − | ik ≤ ≤k REFERENCES now show by induction that ∆ 2√ε 2 5 1 . This k ≤ 5 2 − is trivially true for k = 1, since ∆ = 0. Suppose it is true [1] D. Deutsch, “Quantum theory, the Church-Turing principle and the 1 h i universal quantum computer”, Proc. R. Soc. Lond., vol. A400, 1985. for every 1 i k 1, then in particular, ∆ 1 by the ≤ ≤ − i ≤ 3 [2] R. Feynman, “Simulating physics with computers”, International Journal assumptions on ε given in the statement of this lemma, and of Theoretical Physics, vol. 21, pp. 467-488, 1982. [3] E. Bernstein, U. Vazirani, “Quantum Complexity Theory”, SIAM Journal k 1 − ∆i +2√ε on Computing, vol. 26, pp. 1411-1473, 1997. ∆ [4] J. Gruska, “Quantum Computing”, McGraw–Hill, London, 1999. k ≤ 1 ∆ i=1 i [5] M. Ozawa and H. Nishimura, “Local Transition Functions of Quantum X − Turing Machines”, Theoret. Informatics and Appl., vol. 34, pp. 379-402, k 1 i 3 − 2 5 2000. 2√ε 1 +2√ε ≤ 2 5 2 − [6] P. Benioff, “Models of Quantum Turing Machines”, Fortsch. Phys., vol. i=1 " # ! 46, pp. 423-442, 1998. X [7] J. M. Myers, “Can a Universal Quantum Computer Be Fully Quantum?”, 2 5 k = 2√ε 1 . Phys. Rev. Lett., vol. 78, pp. 18231824, 1997. 5 2 − [8] N. Linden, S. Popescu, “The Halting Problem for Quantum Computers”, " # Preprint, 1998. [Online]. Available: http://arxiv.org/abs/quant-ph/9806054 Thus, it holds that [9] M. Ozawa, “Quantum Turing Machines: Local Transition, Preparation, Measurement, and Halting”, Quantum Communication, Computing, and ek vk ek e˜k Measurement 2, pp. 241-248, 2000. k | i − | ik ≤ k | i − | ik [10] Y. Shi, “Remarks on Universal Quantum Computer”, Phys. Lett. A, vol. + e˜ w + w v 293, pp. 277-282, 2002. k | ki − | kik k | ki − | kik 2 e˜ w + ε [11] T. Miyadera, M. Ohya, “On Halting Process of Quantum Turing Ma- ≤ k | ki − | kik chine”, Open Systems and Information Dynamics, vol. 12 Nr. 3, pp. 261- 2 5 k 264, 2005. 4√ε 1 + ε . [12] M. M¨uller, “Quantum Kolmogorov Complexity and the Quantum Turing ≤ 5 2 − " # Machine”, doctoral thesis, Technical University of Berlin, Berlin, 2007. STRONGLY UNIVERSAL QUANTUM TURING MACHINES AND INVARIANCE OF KOLMOGOROV COMPLEXITY (AUGUST 11, 2007) 18 [13] M. Li and P. Vitanyi, An Introduction to Kolmogorov Complexity and Its Applications, Springer Verlag, 1997. [14] A. Berthiaume, W. Van Dam and S. Laplante, “Quantum Kolmogorov complexity”, J. Comput, System Sci., vol. 63, pp. 201-221, 2001. [15] B. Schumacher, M. D. Westmoreland, “Indeterminate-length quantum coding”, Phys. Rev. A, vol. 64, 042304, 2001. [16] K. Bostr¨om, T. Felbinger, “Lossless quantum data compression and variable-length coding”, Phys. Rev. A., vol. 65, 032313, 2002. [17] C. Rogers, V. Vedral, “The Second Quantized Quantum Turing Ma- chine and Kolmogorov Complexity”, Preprint, 2005. [Online]. Available: http://arxiv.org/abs/quant-ph/0506266 [18] M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. [19] F. Benatti, T. Kr¨uger, M. M¨uller, Ra. Siegmund-Schultze, A. Szkoła, “Entropy and Quantum Kolmogorov Complexity: a Quantum Brudno’s Theorem”, Commun. Math. Phys., vol. 265/2, pp. 437-461, 2006. [20] P. Vit´anyi, “Quantum Kolmogorov complexity based on classical de- scriptions”, IEEE Trans. Infor. Theory, vol. 47/6, pp. 2464-2479, 2001. [21] P. G´acs, “Quantum algorithmic entropy”, J. Phys. A: Math. Gen., vol. 34, pp. 6859-6880, 2001. [22] T. M. Cover, J. A. Thomas, Elements of Information Theory, Wiley Series in Telecommunications, John Wiley & Sons, New York, 1991.