Adiabatic Quantum Transistors

Adiabatic Quantum Transistors

Dave Bacon,1, 2, ∗ Steven T. Flammia,3, 1 and Gregory M. Crosswhite2, † 1Department of Computer Science and Engineering, University of Washington, Seattle, WA 98195 USA 2Department of Physics, University of Washington, Seattle, WA 98195 USA 3School of Physics, University of Sydney, Sydney NSW 2006 Australia We describe a many-body quantum system which can be made to quantum compute by the adiabatic application of a large applied field to the system. Prior to the application of the field quantum information is localized on one boundary of the device, and after the application of the field this information has propagated to the other side of the device with a quantum circuit applied to the information. The applied circuit depends on the many-body Hamiltonian of the material, and the computation takes place in a degenerate ground space with symmetry-protected topological order. Such adiabatic quantum transistors are universal adiabatic quantum computing devices which have the added benefit of being modular. Here we describe this model, provide arguments for why it is an efficient model of quantum computing, and examine these many-body systems in the presence of a noisy environment.

I. INTRODUCTION of one and two-qubit gates enacting a circuit, and fi- nally performs a measurement (readout) of the qubits. The invention of the transistor [1] was a watershed In contrast with modern classical computers where in- moment in the history of computing: it provided a logic formation propagates spatially—advancing roughly one element that was naturally robust to noise and error. step across the computer chip at each rise and fall of a Quantum computers offer the potential to exponentially clock voltage—most proposed implementations of quan- speed up some computational problems (notably factor- tum computers fix the information spatially and bring ing [2]), but have not been built in large part because the computational operations to the data. Many re- quantum information is notoriously fragile and quickly searchers have noted this difference, and a variety of becomes classical information in the presence of noise. In quantum computing (QC) models were subsequently de- theory, the quantum threshold theorem [3–5] asserts that veloped where quantum information propagates spatially these difficulties can be circumvented, but in practice the during a quantum computation; examples include spin- requirements of this theorem are daunting. Here we out- wave models where quantum information moves ballis- line a novel method for building a fault-tolerant quantum tically down a quantum wire [7], linear-optics QC [8], computer that much more closely mimics the classical one-way QC where simple sequential measurements push transistor. In particular, we show how a suitably engi- the quantum information across the system [9], and some neered material can quantum compute by the simple ap- universal adiabatic QC models which create a superposi- plication of an external field to the sample. Applying the tion of computational states spread across the device [10– field adiabatically causes quantum information to spa- 12]. However none of these constructions yield architec- tially propagate across the device at the same time that tures that mimic modern synchronous sequential com- a quantum computation (quantum circuit) is enacted on puter chips. In these chips the rise or fall of a global the quantum information, and this in turn allows us to voltage is the trigger that causes the information in the design clocked quantum computing architectures similar device to move spatially across the device. While the in control requirements to modern classical computers. control requirements for the clock signal in synchronous While we will not be able to rigorously show that our sequential logic devices are by no means trivial, they do adiabatic quantum transistors are fault-tolerant devices not require the precise control, measurement, and tim- arXiv:1207.2769v2 [quant-ph] 18 Jun 2013 like classical transistors, we present analytical and nu- ing that makes quantum computers notoriously difficult merical arguments as to why these transistors could be to build. Here we introduce a new way to mimic these tolerant to errors and thus a true quantum analog of the classical control requirements that allows us to propose classical transistor. synchronous sequential fault-tolerant QC architectures. The standard operating model for a quantum com- The key to our construction is a novel type of quantum puter is called the quantum circuit model [6]. In this gadget that mimics the role classical transistors play in model, one begins with a system of initialized two-level modern computers, and that we hence label an adiabatic quantum systems (qubits), applies a temporal sequence quantum transistor. While the outline we give for an adiabatic quantum transistor is still very far from ex- perimental realization, the novel quantum computational matter that such a device represents opens a new and po- ∗ Current affiliation: Google Inc. tentially promising path for the construction of a large † Current affiliation: Department of Physics, University of Queens- scale quantum computer. land, Brisbane, QLD 4072 Australia An outline of the paper is as follows. In Sec.II we in- 2 troduce a model of an adiabatic quantum transistor. The speed at which we can operate a quantum transistor is re- VGB Vth lated to the minimal energy gap between the ground state manifold and first excited state during the application of Drain n Drain n the applied field. In Sec.IIA we rigorously prove that Gate Body Gate n if the adiabatic quantum transistor enacts a single-qubit Body quantum circuit with only identity gates, then the energy Source n Source n p p gap is such that the adiabatic quantum transistor can be enacted efficiently. In Sec.IIB we provide strong nu- VD = VS VD VS merical evidence from matrix product state simulations ￿ ≈ that the same efficiency holds when the adiabatic quan- tum transistor enacts an arbitrary single-qubit quantum FIG. 1. Graphical depiction of the operation of a classical circuit. Next in Sec.IIC we provide (weaker) numeri- transistor. In a classical transistor such as the n-channel cal evidence that the same results hold for our adiabatic MOSFET shown, the application of a voltage at the gate quantum transistors for quantum circuits involving more input transforms the semiconductor in a manner such that than one qubit. Having given evidence that the adiabatic a channel opens between drain and source. Note that this quantum transistor model is an efficient model of quan- transition is switched by the application of an electric field. tum computation in an ideal world in which the system Traditionally we think of this action as a switch, but there is isolated from its environment, in Sec.III we turn to is another possible interpretation. In particular, the applica- tion of the applied electric field has the effect of copying the the question of whether the model can be made fault- voltage at the source to the drain, which we can interpret as tolerant. We give arguments that if quantum transistors an identity logical gate (a logical gate that takes in a single are configured to execute fault-tolerant quantum circuits bit and outputs this bit unchanged) on the classical infor- then the tolerance of these circuits to errors is conveyed mation encoded as a voltage. Quantum information cannot to our model. While we cannot show that our model has be copied [14], so if we are to imagine a suitable analogy of a threshold theorem associated with it, we can give phys- the transistor in the quantum world, the quantum informa- ical reasons for why the model will have a threshold. In tion must move from the source to drain. This is what our Sec.IV we discuss how the unrealistic four-body Hamil- proposed adiabatic quantum transistors achieve. tonian we use in our constructions can be implemented using a Hamiltonian with only two-body interactions, via perturbation theory gadgets, and discuss the effect that these gadgets have on the arguments in the previous sec- small fault-tolerant quantum circuit for a single encoded tions. In Sec.V we present a variety of constructions for logical gate and be used to classically steer the quantum quantum adiabatic transistors that are likely to be use- information across the device. During the application ful, including systems that can be used to spatially route of the field, the system remains in its ground state, how- quantum information and to perform measurements. Fi- ever the energy gap to the first excited state gets smaller. nally in Sec.VI we conclude and list important open Thus the device functions as an instantiation of a univer- questions for the adiabatic quantum transistor model. sal adiabatic quantum computer [10–12] with, however, the important differences that the ground state of the system can be degenerate (and thus the model could also II. THE ADIABATIC QUANTUM TRANSISTOR be considered an open loop holonomic computation [13]), MODEL and, more importantly, that the information propagates spatially (thus the transistors are modular.) This allows Our proposed adiabatic quantum transistor is a device us to design clocked architectures that very closely re- that operates in a manner similar to a classical logical semble today’s classical synchronous digital computers element such as a MOSFET. See Fig.1 for the analogy (see Fig.3). Here we introduce the details of adiabatic with a classical transistor and Fig.2 for a diagram of how quantum transistors and present theoretical arguments an adiabatic quantum transistor would work under this and evidence from simulations that the energy gap in the analogy. Like a classical transistor this device is made system shrinks sufficiently slowly so as to allow adiabatic to quantum compute via the application of an applied quantum computing. We further present arguments for field to the device. Prior to the application of the field, how our construction can be made fault-tolerant. This the quantum information is localized to one side of the results in an implementation of quantum computers in device, and the system is in its ground state. After the which precision gates, preparations, and measurements application of the field, the quantum information is on are replaced by sufficiently slow and smooth application the opposite side of the device, but with a quantum cir- of fields, and the degree to which one can suitably engi- cuit applied to this information, and the system remains neer a many-body quantum system. in its ground state. The exact circuit applied depends on Our starting point is the recent work that shows how the microscopic details of the system—we propose that it is possible to use piecewise adiabatic deformation of a each such quantum transistor be made to implement a many-body interacting quantum system [15–18] to per- 3

it requires one to precisely control a microscopic applied field that is turned on in a step-by-step process. This mo- tivates the following question: if we take the construction in [16] and, rather than turning on the field in a piece- wise fashion, we instead turn on the field simultaneously across the entire device, then does the device still per- form the desired quantum computation? Here we answer this question affirmatively and show how it leads to our Field adiabatic quantum transistor. The many-body quantum system with the remarkable Time properties given in [16] is described by a twisted version of the Hamiltonian associated with cluster states. Cluster states are the entangled states originally used to perform measurement-based quantum computation [9, 19]. To define the Hamiltonian of our system, we need a graph FIG. 2. Sketch of how an adiabatic quantum transistor be- haves. With no applied field, the quantum information is (with vertices V and edge set E) where each vertex, v, on one side of the device (shaded grey, left) and after the is labeled by an angle θv and is associated with a qubit. adiabatic application of the field the quantum information We define the twisted cluster-state Hamiltonian to be has propagated to the other side of the device (shaded grey, X Y H = cos(θ )[X] + sin(θ )[Y ]
[Z] , right) with a unitary U applied to the original information. C − v v v v w The strength of the applied field is initially zero and then is v∈V (v,w)∈E ramped up to a non-zero value, during which time the quan- (1) tum information is not localized to either end of the device. where [P ]v denotes the operator P acting on the qubit at vertex v, X, Y , and Z are Pauli operators, and we choose units so that the coupling constant is unity. The ground state of this Hamiltonian is a cluster state in a locally rotated frame [16]. Note that the above Hamilto- nian contains interactions that are physically unrealistic Program because they involve more than two qubits. Indeed, the Data Unit State complexity of the interaction increases with the degree Creation Unit of the graph. Fortunately, we need only consider graphs with degree at most 3 (corresponding to four-body inter- Quantum Router actions on a honeycomb lattice graph), and it turns out Logic Unit that we can use perturbation theory gadgets to obtain an effective Hamiltonian of this form inside the low-energy Measurement sector of a Hamiltonian containing only two-qubit inter- Clock Unit actions [15, 20, 21] (see Sec.IV). We will assume here the Hamiltonian in Eq. (1) and return to implementation in two-qubit interactions later. The results in Ref. [16] show that one can take a quan- FIG. 3. Using quantum transistors one can design a syn- tum circuit and, using the recipe shown in Fig.4, map chronous architecture as shown. A router takes classical infor- it to a twisted cluster-state Hamiltonian with the prop- mation about what circuit should be executed (the quantum erty that adiabatically turning on fields along the x di- program) and routes it to the appropriate action unit. These rection ( X on vertex v) and turning off relevant non- units are made up of a collection of quantum transistors that − v can be used to implement their given functions. This infor- commuting terms in HC pushes information through the mation is then directed to the router where another step in device in such a way as to implement the computation de- the circuit begins by reading the next program action. fined by the quantum circuit. In the analysis of Ref. [16] one can separate out the rigorous argument that the com- putation is performed from whether this procedure can be performed adiabatically. Thus, if we consider the case form a quantum computation. In Ref. [16] we consid- of turning on all of the fields simultaneously rather than ered a certain many-body interacting quantum system piecewise, we see that our analysis of the fact that a quan- in its ground state whose Hamiltonian we adiabatically tum computation is performed carries directly over from deformed by turning on a strong field at the border of the Ref. [16]. By contrast, although the energy gap in the device and then slowly propagating this field across the computation is independent of the size of the computa- material; this propagates quantum information through tion in the piecewise construction of [16], when one turns the device in a piecewise fashion, riding the front of the on the field at all locations simultaneously this will no applied field. A major drawback of this method is that longer be true. Therefore, in order to show that turning 4

A. A One Dimensional Wire

Gate Cluster We begin by analyzing the simplest instantiation of our model: n qubits on a line with no twists in the Hamil- + = tonian, θ = 0 for all v V . In this case we evolve the | ￿ ∼ v ∈ θ system according to the time varying evolution H Rθ = n−2 ∼  X  H(t) = f(s) [Z] −1[X] + [Z] [X] +1[Z] +2 θ − − n n i i i H Rθ i=1 n−1 X = g(s) [X] (2) H Rφ ∼ − i φ i=1 where f and g are suitably smoothly varying envelopes which satisfy f(0) = g(1) = 1 and f(1) = g(0) = 1, and t FIG. 4. Dictionary for translating a quantum circuit into s = T is a scaled time. For simplicity, in this subsec- the cluster state Hamiltonian (see one-way QC [9].) First, tion, we will assume f(s) = 1 s = 1 g(s). Note that express the quantum circuit in the universal gate set above we have also turned off the cluster− state− term in Eq. (1) where Rθ = exp(−iθZ/2), H is a Hadamard, and the two- corresponding to the first qubit. This implies that the qubit gate is a controlled-phase gate, along with preparations ground state of the Hamiltonian is two-fold degenerate of the +1 eigenvalue of X, |+i. Then convert this circuit into and thus encodes a qubit [16]. Initially this logical qubit a graph and corresponding twisted cluster-state Hamiltonian is localized on the first two physical qubits of the chain. by converting the circuit graph to twisted cluster state graph using the diagramed dictionary. For each of the unitary oper- At the end of the adiabatic evolution the qubit will be at ators above, the corresponding gadget is labeled by its input the end of the chain on the last qubit, with a Hadamard (shaded) and output (unshaded) qubits. To construct the gate applied to the qubit if the chain has even length. graph from the gate circuit, replace each unitary gate with We can now ask the question, what is the minimum en- the corresponding graph, merging output (unshaded) and in- ergy gap for the above evolution, and how does it scale put (shaded) qubits. For the non-unitary preparation of |+i with the length of the chain? If the energy gap scales the inputs are never merged, but the output is merged with as an inverse polynomial in the length of the chain, then any input that this circuit element proceeds. The end result the adiabatic theorem tells us that propagating the qubit of this construction will be a labeled (by the angles in the Rθ down this length n chain can be done efficiently (i.e., in gates) graph corresponding to a twisted cluster-state Hamil- a time polynomial in n). tonian. Output qubits will not have been merged and will be To answer this question, we use the equivalence of this in the location of the quantum information after the adiabatic procedure described in the text has been completed. model to two uncoupled transverse Ising models [23] (see also [24]). These systems can be exactly diagonalized by a transformation of the spin model to a model with non-interacting fermions. Following the work of Doherty and Bartlett [23], but with additional attention to the boundary terms, suppose that one defines a code with the following (n 1) encoded Pauli operators −  on the field over the entire system still causes our device [X]1[X]3 [X]j if j is odd, X¯j = ··· (3) to execute the desired computation, we must show that [X]2[X]4 [X]j if j is even . the inverse of the energy gap between the ground state ··· and the first excited states grows at most polynomially and in the size of the quantum circuit being enacted. If in-  deed the inverse gap grows this slowly then the adiabatic [Z]j[X]j+1[Z]j+2 if j < n 1, Z¯j = − (4) theorem (for example [22]) guarantees that the system [Z]n−1[X]n if j = n 1 . − will remain in its ground state throughout the evolution conditional on turning on the field over a time scale poly- Then, given these encoded operators, we can then express nomial in the size of the quantum circuit. the Hamiltonian in Eq. (2) as the union of two uncoupled transverse Ising models acting separately on the even and odd qubits, with an additional boundary term: To summarize, quantum circuits correspond, via Fig.4, n−1 n−3 to twisted cluster-state Hamiltonians, and the minimal X  X  H(s) = (1 s) Z¯ s X¯1 + X¯2 + X¯ X¯ +2 . spectral gap of such a Hamiltonian determines how long − − i − i i it takes, via adiabatic deformation, to achieve a high- i=1 i=0 fidelity implementation of the circuit. (5) 5

Using standard techniques for diagonalizing such

Hamiltonians [25] (and the trick of adding an extra qubit 1 to make the system have quadratic fermion operators, Mean and standard deviation Fit of 4.97n-1 see AppendixA), this Hamiltonian can be shown to be Smallest and largest gaps Non-twisted equivalent to a system of non-interacting fermions with

l   )

X 1 6 H (s) = ω (s) η†η l k k k − 2 k=1 2  h kπ
i ω (s) = 4 1 2s(1 s) 1 cos (6) 0.1 k − − − l+1 † where ηk is the creation operator for the kth fermion  n   n+1 

(see AppendixA) and l = 2 or l = 2 depend- Energy gap ( ing on whether one is considering the even or odd chain. Note that there is no k = 0 energy level. From this equation one sees that the minimum energy gap occurs 1 kπ
at s = 2 where ωk(1/2) = 2 cos 2(l+1) and is of order 0.01 1
10 100 O n . Thus, since the minimum gap scales inversely as a Number of qubits (n) polynomial in the length of this one-dimensional system, we see that if we turn the field on in a time polynomial in this length, then with high probability the quantum FIG. 5. Energy gap versus number of qubits for a one- information will propagate from one end of the system to dimensional twisted Hamiltonian with random angles. The the other end—with a possible Hadamard gate applied, duality condition has been enforced so that the minimum is 1 depending on whether n is odd or even. at s = 2 (see text.) We took 200 samples with uniformly dis- tributed random angles. We also plot the smallest and largest energy gaps that were obtained for a given number of qubits B. Single Qubit Quantum Circuits (dotted lines) as well as the case with no rotated angles (which we showed could be computed exactly) (non-bold line.) The data are well fit by (4.97 ± 0.02)n−1. Having demonstrated rigorously that the scaling for a quantum wire has a gap which scales inversely with the length of the wire, we now examine what happens when we apply other single qubit gates by using twisted Hamil- Define the following unitary operator: tonians with varying θ’s. Here we give strong numerical evidence that the gap scales inversely as a polynomial n ! n−1 ! n−2 ! Y Y Y through the use of a matrix product state algorithm. To U = S [Z]1 [X] [C ] +1 [R( θ )] +1 i Z i,i − i i simplify our study of the one dimensional twisted Hamil- i=2 i=1 i=1 tonian, it is convenient to note that this model has a (9) duality. In particular, for a twisted Hamiltonian with where R(θ) = exp( iZθ/2), [C ] is the controlled- angles θi where 1 i n 2 and θi is the angle associ- − Z i,j ated with the (i +≤ 1)st≤ qubit,− it can be shown that if the phase gate acting between qubits i and j, and S is the angles obey the symmetry condition θi = θn−i−1 for all i gate which inverts the chain about its middle, swapping the 1st and nth, 2nd and (n 1)st, etc. qubits. Then one and additionally there is only one minimum in this model − (which is true numerically for small systems), then the can check that minimum energy gap occurs at the midpoint, s = 1/2. n−2 † X To see this, we proceed as follows. First we define a UH(s)U = s [Z] [θ − −1] +1[Z] +2 − i n i i i shorthand for the operators which are rotated combina- i=1 tions of X and Y Pauli operators, namely, n−1  X [Z]n−1[X]n (1 s) [X]i. (10) [θk]i = cos(θk)[X]i + sin(θk)[Y ]i . (7) − − − i=1 Now consider the twisted one-dimensional cluster Hamil- tonian with a transverse field, Thus, in the case that θi = θn−i−1 for 1 i n 2, H(s) has the same spectrum as H(1 s) and≤ in≤ particular− if n−2 ! − X there is only one quantum phase transition (meaning, H(s) = (1 s) [Z]i[θi]i+1[Z]i+2 + [Z]n−1[X]n only one value of s for which the spectral gap above the − − i=1 degenerate ground space collapses to zero in the thermo- n−1 dynamic limit), then it occurs when s = 1 s or s = 1 . X − 2 s [X]i. (8) Given this, we used the technique of matrix product − i=1 states [26–28] to calculate the energy gap for systems 6 having up to 200 qubits with angles θi chosen uniformly If one instead applies Uy, consisting of controlled-phase from [0, 2π) satisfying the aforementioned symmetry con- gates between all y-coordinate neighbors, [CZ ](i,j),(i,j+1) dition; for each system size that was examined, we per- to the Hamiltonian in Eq. (11), then we see that this formed 200 simulation runs, each with a different choice simply swaps the direction of the stripes: of angles. Fig.5 presents the resulting data, from which L we find that the energy gap data are well fit by a function X U HU † = (1 s)[θ ] [Z] [Z] which scales as O(n−1), thus indicating that the twisting y y − − (i,j) (i,j) (i,j−1) (i,j+1) =1 of the Hamiltonian does not quantitatively change the i,j scaling of the energy gap with the length of the single + s[Z](i−1,j)[X](i,j)[Z](i+1,j) . (13) qubit circuit being implemented. If one swaps the qubits about the line x = y in Eq. (13), we obtain the striped Hamiltonian in Eq. (12), but with θ replaced by θ and s replaced by 1 s. Thus if we C. Multi-qubit Quantum Circuits (i,j) (j,i) − enforce θ(i,j) = θ(j,i) we obtain a duality, and as in the previous subsection there will be a minimum at s = 1 . Finally we turn to the question of the energy gap in the 2 For systems that obey the duality condition θ = case where the circuits involve two or more qubits. Here (i,j) θ we have investigated the size of the energy gap using we note that if we take a square lattice with appropriate (j,i) exact diagonalization as well as a matrix product state boundary conditions, then with the untwisted Hamilto- approach. The results of these simulations are plotted in nian model is equivalent to a quantum compass model Fig6. While we can only obtain weaker evidence of an restricted to a certain symmetry sector. Under this sym- inverse polynomial for this two-dimensional system, the metry restriction, the minimal gap occurs at s = 1 , 2 data are at least consistent with this hypothesis. where the compass model has equally competing inter- actions. Numerical evidence from exact diagonalization on a square lattice for this model presented in Ref. [29] III. FAULT-TOLERANCE OF THE ADIABATIC demonstrates that the minimal energy gap for this model QUANTUM TRANSISTOR with the symmetry restriction scales as 1/n where n is the size of (i.e., number of qubits in) the square lattice. Having given strong evidence that our model can effi- Thus the real question is what happens to the energy gap ciently produce a desired quantum circuit when all evo- for more general circuits. lutions are error free, we now turn to the question of We consider a model on a square lattice with the full how this model will behave in realistic settings where twisted cluster state Hamiltonian turned on interpolating the system is coupled to an environment. Here we argue to a Hamiltonian with applied fields on across the entire that if one uses standard circuit constructions for fault- device. Label the qubits on the square grid by (i, j). The tolerant QC [3–5], this will be good enough to ensure ro- Hamiltonian we consider is bust quantum computation in the model. In particular L we can consider a model in which each modular adiabatic X H =(1 s)H s [X] (11) quantum transistor is configured to perform an encoded − C − (i,j) i,j=1 quantum gate, state preparation, or measurement from a fault-tolerant quantum circuit construction, including where L is the length of the lattice and we define [P ](i,j)s the error correcting step. We will argue that if one does with values of i or j lying outside the lattice as identity this, then the standard analysis of the success probabil- operators to deal with boundary terms. Here, HC is the ity of such constructions carries over to our model. Note twisted cluster Hamiltonian of Eq. (1) for a square lattice that fault-tolerant QC requires expunging entropy (usu- with some particular choice of angles θ(i,j) for each vertex ally via measurement), but this can always be placed at (i, j) in the lattice. We suppress the dependence on the the end of the fault-tolerant block [30]. Thus we envision θ(i,j) to avoid notational clutter. here a model in which the final measurements in error If we apply a unitary operator Ux consisting of a correcting circuits are implemented after each evolution controlled-phase gate between all x-coordinate neighbors, of a quantum transistor. This can be done by either di- [CZ ](i,j),(i+1,j), to the Hamiltonian in Eq. (11), then the rectly measuring the relevant qubits, or via an adiabatic cluster state Hamiltonian is turned into a y-direction amplifier that dissipates energy by the natural relaxation striped twisted Hamiltonian and the applied field turns of the system, as described in Sec.VA. into a x-direction striped untwisted Hamiltonian. Ex- First note the following positive result about using each plicitly, this transformed Hamiltonian (which will have quantum transistor as a fault-tolerant circuit gadget. If the same spectrum) is given by one is executing a quantum algorithm with L gates, then

L each of these gates needs to be executed with accuracy X −1 U HU † = (1 s)[θ ] [Z] [Z]  = 0L . To do this using fault-tolerant gates below x x − − (i,j) (i,j) (i+1,j) (i−1,j) the fault-tolerant threshold requires [3–5] that the gad- i,j=1 gets have circuits of size Ologc −1
for a constant c. + s[Z](i,j−1)[X](i,j)[Z](i,j+1) . (12) Thus if the energy gap in an adiabatic gate shrinks as 7

a universal set, and indeed our results in Fig.6 already 1 constitute some measure of support for this statement. Untwisted Random angles -1 Proving a gap for a universal set of logical gates remains Fit of 1.63n an important open problem related to this work. )

6 A. An Unrealistic But Illuminating Error Model

We now turn to the question of whether fault-tolerant 0.1 gadgets will act to overcome errors in the adiabatic tran- sistor. We begin by considering the following simplified error model: assume that at the beginning of the com- putation the system is not in the ground state of H , Energy gap ( C but rather it is in one of the excited states of HC . Note that since the terms of HC commute (as expressed in Eq.1), excited states of HC can be labeled by the list

0.01 of eigenvalues of each of the terms, where the eigenvalue 10 of each term is either +1 or 1. If one carries out the Number of qubits (n) analysis described in Ref. [16−] regarding the computa- tion that is performed if one starts in such an excited state, then one sees that for each of the terms in HC FIG. 6. Energy gap versus number of qubits for the two- whose initial eigenvalue is +1 there is a corresponding dimensional twisted Hamiltonian with random angles and the Pauli error in the circuit—in other words, errors that duality condition enforced (see text.) The initial Hamilto- take the form of starting in an excited state of H map nian was a rotated cluster Hamiltonian for a square lattice C (Eq. (1)) and the final Hamiltonian was an X field over the directly onto errors in the circuit model. For example, entire device. All data were obtained using exact diagonal- consider the one-dimensional case with no rotated an- ization with accuracy 10−8, with the exception of the random gles; if the initial state is in the +1 eigenspace of some angle data for n = 25 which were obtained using a one di- term [Z]i−1[X]i[Z]i+1 in the middle of the chain, then − mensional matrix product algorithm wrapped onto the two- this corresponds to a Pauli Z error in the circuit model. dimensional lattice and have accuracy 10−3. The data are Furthermore, note that in this model errors that are −1 well fit by (1.63 ± 0.12)n . We also show the untwisted en- local on the physical qubits of HC map to errors that are ergy gap for comparison. The untwisted energy gap does not local in the circuit since all terms in HC are localized on a appear to be a power law because there is an even-odd lat- few qubits—so for example, applying [X]i to the system tice effect: examination of non-square lattices gives us more flips only two eigenvalues of H in the untwisted case. By confidence that the untwisted case is indeed a power law. C the linearity of quantum mechanics, we can extend this argument to initial preparations that are mixtures and coherent superpositions of such errors. Thus if we take our initial system and expose it to an environment for an an inverse polynomial, the energy gap in the individual amount of time proportional to the size of the quantum k −1
fault-tolerant gadget will shrink as O (log L) for a circuit we are about to enact, then subsequent perfect constant k. Because each individual transistor executes adiabatic evolution will produce errors in the quantum a single encoded gate, this means that each adiabatic circuit model that are commensurate with a local inde- evolution need only be polylogarithmically longer when pendent error model. (Note that this model does have executing larger and larger quantum algorithms. temporally correlated errors over a few qubits, but this Second, and equally important, is the fact that a spec- does not affect the existence of fault-tolerant methods for tral gap need only hold for some universal set of encoded overcoming these errors [3].) gates. In other words, we do not need to have a spectral gap for all possible Hamiltonians of the type in Eq. 12, we only need a gap for some particular set of logical gates. B. Errors On A Quantum Wire For example, if one could prove that logical Hadamard, CPHASE and π/8 gates have an appropriately large gap, Of course, this model in which errors appear only at the that would be sufficient for any algorithm in our scheme beginning of the computation is a fiction. More generally to have a sufficiently large gap, since it would be com- we may consider a model in which errors occur through- prised of solely these gates. Having a gap for only a hand- out the adiabatic evolution. Consider such an evolution ful of specific models is a tremendously weaker require- for the single quantum wire with an untwisted Hamilto- ment than having a gap for all twisted cluster Hamilto- nian. As we detailed above, this model can be mapped nians. We expect that, at the very least, substantial nu- into two transverse-field Ising models. Any linear opera- merical evidence could be gathered in support of a gap for tor, and hence any error, on this model can be expressed 8 as one of three types of errors: errors that change the energy of the system, errors that act on the degeneracy (corresponding to the encoded qubit), and a combination 2.0 of these two errors. We will show that each of the first two types of errors can be mapped to an independent error 1.5 model on the quantum circuit corresponding to the quan- tum wire in an independent manner, thus taking care of 1.0 the third type of error. Finally note that we use the term Energy error in a quantum error correcting sense to denote an 0.5 error operator, even when this arises from coherent error sources. For example our results deal with the errors in 0.0 the Hamiltonian description resulting from a perturbing interaction on the system H0 = H + λV for small λ as 0.0 0.2 0.4 0.6 0.8 1.0 well as for couplings of our system to an environment, s though the argument for these errors is slightly different (see Sec. IIIB3 for a discussion of this point.) FIG. 7. When one exactly solves the one-dimensional wire, the result is that the wire behaves like a collection of free 1. Errors that change the energy of the system fermions. Here we show the energies for creating each of these fermions in a chain of length 80 (thus all the energy levels are sums of these energies) as a function of the scaled time s. We first consider errors that change the energy of the See Eq. (5) in AppendixA. For high energy excitations, if system. The energy spectrum calculated in Sec.IIA has the environment is cool enough, then these energy levels are low-lying excited states and thus we can reasonably ex- essentially never excited (solid lines.) At energies comparable pect excitations into these levels. Assuming detailed bal- to the temperature of the environment, errors will no longer ance in our error rates (which tends to be the case for be suppressed by a Boltzmann factor (dashed lines.) At the weak coupling errors that change the energy of a sys- lowest energies, the interaction strength between the environ- tem [31]) we consider a model in which the rate of errors ment and the system is small enough that the error model ∆E will not be governed by detailed balance. However, assuming − k T that change energy by ∆E scales as p0e B where p0 is that these energy levels are changed by any process whatso- a bare error rate, kB is the Boltzmann constant, and T ever is no worse than our assumption that all errors below is the temperature of the environment. the temperature cutoff lead to real errors. Consider working at a temperature much less than the bare energy gap of the system at s = 0, which is ∆ = 2. In this case the only fermionic excitations that will occur with high probability are those for which ω(s) < kBT . high energy excitations are suppressed by a Boltzmann This condition implies that only a constant fraction of factor, we see that errors in the quantum wire that change fermions will be created in such an error model through- the energy of the system result in independent errors on out the adiabatic evolution. Furthermore, note that be- the quantum circuit being enacted. cause the adiabatic evolution preserves the energy levels, it will therefore proceed to drag the system along the excited energy level. We see, therefore, that errors that 2. Errors on encoded quantum information excite a constant fraction of fermions can be mapped back to excitations at the beginning of the adiabatic evo- The situation for errors that do not change energy but lution; as we have already shown, this is not a problem instead act on the degeneracy of the quantum wire is for standard quantum-error-correction procedures. slightly more complex. To see what happens, we con- Finally we note a technical condition: we have assumed sider the two logical operators encoded in the degenerate a detailed balance error model, but for the lowest lying ground state of the quantum wire, which for a chain of energy levels, where the strength of the interaction be- odd length n are given by tween the system and environment is greater than the en- n+1 n−1 ergy gap, this condition might not hold. However, these 2 2 energy levels are, under our temperature assumption, al- Y Y X¯n = [X]2i−1 and Z¯n = [Z]n [X]2i. (14) ready assumed to be in error, and therefore covered by i=1 i=1 our argument (i.e., they are at most a constant fraction of the error, the fraction being directly related to the ratio The first n 1 operators are the encoded operators of the of the energy gap and the perturbation energy strength.) code described− in Sec.IIA. Note that these two logical See Fig.7 for a graphic depicting the relevant energy operators commute with the entire Hamiltonian during levels and the partition of these errors into different cat- the adiabatic evolution. At the beginning (end) of the egories in our argument. computation this information is localized onto the start To summarize: under the reasonable assumption that (end) of the quantum wire. Thus during both of these 9 times the quantum information in the wire is susceptible Thus using correlation functions for the relevant to decoherence from local errors. For example undesired degeneracy-preserving operation we see that the ampli- or imprecisely controlled terms in our initial Hamiltonian tude for the degeneracy-preserving error is exponentially will be able to act non-trivially on the initial quantum small (as a function of n) except for s near s = 1 in a − 1 − information. However, during the middle of the compu- window of size O(n 2 ) for  > 0. In and of itself this tation, we will argue that the information in the degen- would imply that the model is in trouble: with a linear eracy is protected, and furthermore that we can adjust interpolation for this would imply spending O(√n) time the adiabatic schedule in such a way that the system only in this region during which the quantum information can spends a constant amount of time during which the quan- be decohered. However, note that the energy gap for this tum information is exposed to local decoherence at the model is large during the beginning and end of the adia- beginning and end of the adiabatic evolution. batic path. Thus instead of using a linear interpolation ¯ In particular consider the logical Xn. By expanding as in Eq. (2) one can use an interpolation which spends this logical operator in terms of the underlying physical only a constant amount of time in the window where operators, we see that this error requires a combination these errors can affect the logical quantum information. of O(n) local [X]i errors on odd qubits. When we map In particular if we maintain the adiabatic condition these errors over to the transverse Ising model, these are for each infinitesimal evolution of the system we can ob- errors on the Ising model, but are now two-qubit terms tain a schedule for interpolating our Hamiltonian which ¯ ¯ ¯ like XiXi+1 for i 2 or X1. Individually these errors can spends significantly more time where the energy gap is ≥ change the energy of the system (and we have previously smaller. In particular, consider the adiabatic condition argued that such errors can be dealt with), but they can for the one-dimensional quantum wire. As we show in also have an effect which keeps the energy constant. To the appendix, the energy spectrum for a wire of length evaluate this effect, for example, for information at the 2l is given by Eq. (A22). To maintain the adiabatic con- end of the computation, we can evaluate the portion of dition for the lowest energy level infinitesimally we must the error amplitude which preserves the vacuum of the n satisfy [33] Q 2 ¯ ¯ transverse Ising model: e = i=0 v XiXi+1 v where v h | | i | i ds 2 ds is the vacuum (the single X¯1 error can be thought of as = ωl (s) dt = 2 , (19) a two-qubit X error on the fictitious extra qubit.) dt ⇔ ωl (s) For the transverse Ising model with periodic bound- where  is the accuracy that we desire for our adiabatic ary conditions in the thermodynamic limit, the nearest evolution relative to the ideal evolution. Performing this neighbor XX correlations are given by [32] integration and enforcing t = 0 at s = 0 yields the solu- Z π tion 1 −1   v [X] [X] +1 v = Ω (k) cos(k) + α dk , (15) i i π 1  π  h | | i 0 t = csc 4 l + 1 where   πl −1 h π
i p + tan (2s 1) cot . (20) Ω(k) = 1 + α2 + 2α cos k (16) 2(l + 1) − 2(l+1) s Inverting this equation for s yields a schedule for adjust- and, to relate back to the quantum wire model, α = 1−s . This can be transformed into ing s as a function of time which runs for a total time of 1   v [X]i[X]i+1 v = (2s 1)K 4s(1 s) πl  π  h | | i πs − − T = csc (21) 1 4(l + 1) l + 1 + E4s(1 s) , (17) πs − which scales linearly as l for large l (thus there is no where K and E are the complete elliptic integrals of the Grover-type speedup using this schedule.) The amount of first and second kind respectively. We are interested time during which this schedule spends with 1 η s 1 − ≤ ≤ in the amplitude for the product of n of these correla- is n/2 tion functions, v [X]i[X]i+1 v . Series expanding this T  2(l + 1) h i about s = 1, weh find| | i t = 1 + tan−1 (2η 1) cot π
. η 2 πl − 2(l+1) n/2 1 2 1 3 (22) v [X] [X] +1 v = 1 n(s 1) + n(s 1) h | i i | i − 8 − 4 −  2  For the case relevant to the degenerate error model where 2n 53n 4 −1/2 1 + − (s 1) η = l this is bounded by  . Thus this schedule 128 − spends only a constant amount of time in the region + O(s 1)5
. (18) where the logical output quantum information is suscep- − tible to noise. − 1 −
If (1 s) < O n 2 for  > 0, we see that this function In conclusion, for the single-qubit quantum wire case, vanishes− as n . we see that the quantum information in the degeneracy → ∞ 10 is vulnerable to degeneracy-preserving errors only at the known to be stable [37, 38]. (See also Refs. [39, 40] for beginning or end of the computation, and in a setting other schemes for quantum computation in a symmetry- that is no worse than the standard circuit model with protected topologically ordered phase.) independent errors. We can adjust the adiabatic sched- Next we turn to the case in which there is a static ule so as to not spend much time in these regions. Ac- perturbation, but where we now concern ourselves with tually, this is no different than how we would want to the effect this has coming from portions of V that do operate a quantum computer with or without adiabatic not act on the degeneracy. These perturbations have the quantum transistors: to keep the effective error-corrected effect of changing the energy levels, in terms of both the error rate down, one needs to spend a minimal amount eigenvalues and the eigenvectors of the system. of time letting the quantum information simply decohere. Consider first the effect of the perturbation on the In our model this means that we need to move faster than eigenvectors of the system. We begin by recalling the ar- linearly out of the region where the encoded quantum in- gument that shows that the quantum wire Hamiltonian formation is exposed to errors. We note that the above correctly transmits information in the absence of a static analysis works for both open system errors and also for perturbation. Recall that the quantum wire Hamiltonian (small) deviations in the Hamiltonian we implement, i.e., is given by perturbative system errors. n−2  X  H(t) = f(s) ∆ [Z] [X] +1[Z] +2 ∆[Z] −1[X] − i i i − n n 3. Static Errors i=1 n−1 X g(s)∆ [X] . (23) In the previous subsection we argued that an adiabatic − i i=1 quantum transistor configured to act as a quantum wire would behave, given an appropriate adiabatic schedule, Initially we can define the logical operators X¯ = [X]1[Z]2 as a quantum circuit with an independent error model. and Z¯ = [Z]1 and the system is in the +1 eigenstate of In this argument we have explicitly used an argument the vertex operators [Z]i[X]i+1[Z]i+2 (1 i n 2) that relied on detailed balance. Thus one might wonder ≤ ≤ − and [Z]n−1[X]n. By suitable multiplication of the en- how the argument works when there is no environment coded logical operators with these vertex operators, we and the errors occur solely from a static perturbation on can express the logical operators as a pattern of [X]i the wire. Here we argue that these errors also give rise to operators on qubits, for i < n, and a Pauli operator   an independent error model, at least for a quantum wire ¯ Qn−1 ¯ with no gates. on the last qubit: X i=1,i odd[X]i [X]n, Z   ≡ ≡ In particular consider a model in which a static term Qn−1 i=1,i even[X]i [Z]n, where we have assumed for sim- is added to our adiabatic evolution H0(s) = H(s) + λV plicity that n is odd. At the end of the evolution we will for the case of a quantum wire. We will assume that V is be in the +1 eigenstate of all of the [X] , 1 i n 1. a sum of local operators (for example, one- or two-qubit i This result then implies that the information≤ will≤ be− cor- operators) and that the strength of the interaction is per- rectly transmitted down the quantum wire since under turbative (λ 1, since we have fixed to unity the energy this condition the logical operators are just bare Pauli scale used in the twisted cluster-state Hamiltonian and operators on the last qubit. the final applied field). First note that the contribution Suppose, on the other hand, that we are initially in a of V to splitting the degenerate ground state of the wire state in which we are in the 1 eigenvalue eigenspace of a will be exponentially suppressed except at the beginning single vertex operator. Then− the above argument about and end of the adiabatic evolution, as we have argued the logical operators could still be applied, but the ef- above. The reason for this is exactly the same: local fect would be that one of the logical operators might operators do not act on this space except when multi- acquire a phase. For example, it might end up that X¯ plied together (except at the beginning and end of the     ≡ Qn−1 [X] [X] , Z¯ Qn−1 [X] [Z] , adiabatic evolution). Thus a term like λV will act like a − i=1,i odd i n ≡ i=1,i even i n single-qubit error term only for the time we spend with which represents the quantum wire being faithful up to the information localized on either end of the wire. Us- a Z error. Generalizing, we see that initial states that ing the scheduling trick we have described previously, are not in the +1 eigenstates translate into single-qubit this means that it acts like an independent error (from a errors on the wire (so that multiple vertex operators in static term) at the beginning and end of the computation 1 eigenstates represent multiple errors.) A similar ar- represented in the wire. Thus we see that the static per- gument− applies to the end of the evolution: if we are are turbing term will not act on the degeneracy in a manner in 1 eigenstates of some final [X]i operator, this will that is inconsistent with an independent error model. correspond− to a single-qubit error on the wire. Note that Moreover, many types of errors will hardly effect the it is possible for the errors in our model to cancel out. degeneracy at all since it is known [34] that the cluster How does the static perturbation changing the eigen- state has symmetry-protected topological order [35, 36] values affect this argument? Consider first the effect of a and the degeneracy and spectral gap for such systems are perturbation on the initial ground state and expand its 11 effect using perturbation theory. In particular if ψg(0) Λ￿0 is the initial ground state of the unperturbed system| andi 1.0 0 ψg(0) is the initial ground state of the perturbed sys- |tem, theni the first-order correction will be 0.8 0.6 0 X ψk(0) V ψg(0) ψg(0) ψg(0) + λ h | | i ψk(0) | i ≈ | i Eg Ek | i k6=g − Energy 0.4 + O(λ2) (24) 0.2 where ψk(0) are the excited energy states of the unper- 0.0 turbed| system.i Now notice that if V is a sum of local 0.0 0.2 0.4 0.6 0.8 1.0 terms, each local term will act to change the eigenval- s ues of at most three vertex operators. Thus, the sum above will not be over all excited states, but will only Λ￿0.1 include excited states with at most three vertex opera- 1.2 tor eigenvalues flipped. Each of these terms represents 1.0 errors that are localized in space-time on the quantum circuit version of the quantum wire, as per our argument 0.8 in the previous subsection. If we were to end up in the 0.6

+1 eigenstate of the final Hamiltonian’s [X]i terms, these Energy terms would then each represent a wire in which at most 0.4 three single-qubit errors occurred during the evolution. 0.2 In other words, the portion of the wavefunction arising 0.0 to first order in perturbation theory, if dragged to the 0.0 0.2 0.4 0.6 0.8 1.0 +1 eigenstates of the [X]i operators, acts as if one had s (nearly) independently erred the quantum information transmitted through the quantum circuit. A similar ar- FIG. 8. Static perturbations. Consider a quantum wire with gument can be made for the effect of being in the wrong repeated Hadamard gates, but where the individual interac- final state. Thus we see that the effect of being in the tions in, for example Eq. (23) are randomly perturbed by a wrong initial and final eigenspace is, for a weak local per- random amount in the range of −λ to λ. This plot is done turbation, equivalent to an error that is local in space and for a wire of length 20. This model can be mapped to the time in the quantum circuit. transverse Ising model and above we plot the energy of the Having shown that the effects of changing the eigenvec- free fermions that occur in this model. One sees that while tors of the ground state can be modeled as an indepen- the energy cost for each fermion with the perturbed field is modified, qualitatively the cost for each individual fermion dent error model on the quantum wire, we now turn to remains approximately the same. This result implies that the the effect of static perturbations on the eigenvalues. The model will behave similarly to the unperturbed model: in the worry here is as follows: the effect of the perturbation absence of other errors the worst this can do is cause a level may cause the energy gap to close and therefore the adi- crossing near s = 1/2 which will at most introduce a constant abatic evolution will actually cause the evolution to not number of errors in the model. Furthermore, the effects of preserve the ground state (subspace). This will certainly other errors occurring on this perturbed model will be similar happen in the system, but the real question is whether to how we argue they will act in the main text. In particu- this looks like an independent error model or not. lar, there will be a constant fraction of errors occurring when To see that this is unlikely to be a problem, consider a the system is coupled to a thermal bath and obeys detailed model in which each of the individual terms in the Hamil- balance. tonian of the quantum wire, Eq. (23) is randomly per- turbed by λ. When we convert this model into the free- fermion model± as discussed in previous sections, we see imagine sweeping adiabatically from s = 0 to s = 1. The that this will result in A and B matrices from Eq. (A3) energy levels that will cause problems are then the free that are like the band diagonal matrices in Eqs. (A5) and fermions with excitation energies below λ. At all stages (A6), but now with random diagonal and off-diagonal ele- where the fermion energies are greater than λ there is a ments added to these matrices. One can then numerically cost to create such a fermion that is greater than λ, and diagonalize the equation for the free fermion energies. In the perturbing potential V will not be strong enough to Fig.8, we show both the unperturbed free-fermion ener- achieve this transition. Since only a constant fraction of gies and also the perturbed energies for a chain of length the energy levels cross below the λ creation energy line 20 and a perturbation strength of λ = 0.1. From this one (similar to Fig.7), this means that the possible errors sees that while the perturbation causes the energies asso- created are a constant fraction of those that could occur ciated with the free fermions to change by λ, the spec- on the quantum wire. This, then, is nothing more than trum is qualitatively the same. In particular,± we can now an independent error model on the system. Thus, at 12 least for the kind of error model we have assumed, static ergy gap in the ancillas of these constructions, one can perturbations result in an independent error model. effectively treat the errors arising from these gadgets as We have argued (but not rigorously proven) in this local errors in our above model. In particular, as in prior subsection that the effect of a static perturbation on a work [15, 20, 21], we can treat the imperfections aris- quantum wire is no different from an independent error ing from the use of perturbation gadgets as terms that model on the quantum circuit corresponding to this wire. cause errors on our system. Thus the only major quanti- tative change resulting from using these gadgets is that the strengths of the engineered interactions will be lower 4. Summary because of the perturbative nature of these gadgets. The best possible choice of gadget will depend on the physical In this section we have argued that errors on quantum system being used and on the relevant noise sources that adiabatic transistors configured as a quantum wire act are present. similarly to errors within the standard quantum circuit The main question that arises from the use of pertur- model with independent (or slightly time-correlated) er- bation theory gadgets is whether they will cause extra rors. Errors that change energy were argued to only act problems when the quantum transistor is configured to as a constant fraction of independent errors in the quan- carry out a fault-tolerant quantum circuit element. In tum circuit model. Errors that preserved the degeneracy the supplementary section of Ref. [15], we analyzed the could be severe, but only during the initial and final du- effects of perturbation theory gadgets for implementing a rations of the adiabatic evolution when the quantum in- simple adiabatic scheme that performs a two-qubit gate. formation is localized on either end of the quantum wire. The main result of this analysis was that the use of per- By spending less time in the adiabatic evolution during turbation theory gadgets to achieve these many-body in- these stages, we showed that these errors act as constant teractions does not substantially damage the usefulness rate errors at the beginning and end of the computation. of these gadgets in the adiabatic constructions. However, Finally, we analyzed how static (not varying in time) per- the use of gadgets has two drawbacks: (1) the eigenvec- turbative interactions would modify these arguments and tors are slightly changed because of the inexact nature of gave evidence that these errors behave in a similar man- the effective many-qubit interaction, and (2) the eigen- ner. Thus, adiabatic quantum transistors configured as values are also slightly changed and thus the gap is made quantum wires seem to have errors that act as indepen- slightly smaller. For the case analyzed in Ref. [15], there dent single-qubit errors in the standard quantum circuit were two energy scales, λ and ∆. We can choose units model. The case of a full quantum circuit with multi- where ∆ = 1. The effective three-qubit interaction in ple qubits is more difficult to analytically or numerically the scheme is of strength O(λ2), and thus the gap in this analyze. Some of our arguments, for instance the effect construction is of this order. The imperfections in the of local perturbations on the initial twisted cluster-state gadget construction led to corrections to the energy of or- Hamiltonian that are local in the space-time of the quan- der O(λ3). Furthermore, the eigenvectors of the ground tum circuit enacted by the adiabatic evolution, carry over state were corrected in amplitude to order O(λ2). The directly to this more complicated case. The full general main conclusion of this calculation was that the effects case is an open question for future work. of the perturbation theory gadgets were small in the per- turbation parameter λ. What can we say about the similar calculation for the IV. PERTURBATION THEORY GADGETS adiabatic quantum transistor? The first question is how the gadgets will affect the initial and final eigenvectors Finally let us mention issues arising from the use of of the system. Bartlett and Rudolph [20] considered ex- perturbation theory gadgets in implementing the many- actly this problem for their encoded gadget scheme, and qubit interactions in HC . Perturbation gadgets pro- showed that the new eigenvectors behave as if there is an duce an effective, low-energy many-qubit interaction via independent error model acting on the state when it is strong and weak single- and two-qubit interactions. The used for measurement-based quantum computing. One strength of these interactions is related to the ratio of can perform exactly the same sort of calculation for the the weak and strong interactions (hence the name.) For perturbation theory gadgets that use ancillas as media- example, one can engineer an effective three-body inter- tors and the conclusion is exactly the same: the lowest- action at second-order in perturbation theory by using order errors created using these gadgets change the state only physically realistic two-body interactions. to act as if it has been independently erred. The result If one wishes to only apply fields (single-qubit inter- is not dissimilar to that of our Eq. (24), but now for the actions) to our model, it appears that one cannot use more complicated quantum transistor with a quantum Bartlett-Rudolph gadgets [20], but instead must use gad- circuit. The other question is what effect the gadgets gets that do not use encoded quantum information [21] have on the energy gap. We note here that the correc- (this is because the former gadgets have encoded quan- tions to the effective interaction for the perturbation the- tum information that is not acted upon by single-qubit ory gadgets can be thought of as extra static Hamiltonian operations.) At temperatures much lower than the en- terms that are weaker by the perturbative factor used to 13 create the gadgets, hence these errors are covered by the argument in the previous section. The use of perturbation theory gadgets in building adi- Gate Cluster abatic quantum transistors is clearly one of the largest drawbacks in our scheme and one of the reasons why we consider this work as an outline for how an adiabatic quantum transistor could work. Recent work using, for example, more physically realistic AKLT states [39–43], X = however, indicates that these gadgets may not be a nec- ∼ essary component of a quantum transistor construction.

V. BUILDING BLOCKS FOR ADIABATIC QUANTUM TRANSISTOR ARCHITECTURES FIG. 9. An adiabatic measurement amplifier. A binary tree with alternating degree 2 and 3 internal vertices (as shown) Finally, let us give details about some building blocks can be used to take information encoded into the X¯ eigen- for adiabatic quantum transistors that will be useful for states at the root into a code across the leaves by turning on building a larger quantum computer architecture that an X field across the shaded qubits. This code has a stabi- closely mimics today’s modern clocked classical computer lizer given by ZiZj terms on leaves; i.e., it is the repetition architectures. These include constructions for a system code. The logical information initially at the root in the X that performs measurements and for a system that can eigenstate is in this process copied into the Z eigenstates of P be used to take classical information and use it to route the leaves. Thus if one turns on a i Zi interaction, the in- quantum information in a device built with an adiabatic formation initially in the X eigenstate will be separated in quantum transistor. The former is essential for fault- energy by twice the number of leaves. The natural relaxation of this system to its ground state will result in O(# leaves) re- tolerant constructions, which must purge the entropy of laxation events, which can be individually (noisily) observed, quantum errors, and the latter is important for building hence amplifying a measurement in the X eigenvector basis. architectures that are modular and synchronous.

multiply the Z¯ operator by graph stabilizers from ver- A. Adiabatic measurement amplifier tices of degree 3 and the graph stabilizer from the leaves, we obtain an operator that has X or I on the non-leaf Here we describe the details of an adiabatic measure- vertices, and is X on all of the leaves. Similarly, if we ment amplifier. This amplifier can be used to take quan- multiply a X¯ operator by a graph stabilizer from vertices tum information encoded on a single quantum wire (in of degree 2 on any simple path from a root to a leaf we the circuit being adiabatically simulated) and copy it to will obtain X operators on the non-leaves, and a Z on multiple qubits in a fixed basis and then read out this the leaf. This latter fact is independent of which path to information. Consider a tree with alternating internal the leaf one chooses, which is a symptom of the fact that levels of degree 2 and 3 as shown in Fig.9. Call the the information is encoded into an error-correcting code. degree-1 external node that is connected to an internal Indeed it is not hard to see that by multiplying graph degree-3 node the root of the tree and label it r. Label stabilizer elements one can obtain ZiZj terms that act the unique child node of r by r0, and let all other degree- only on i and j which are leaves of the graph. These are 1 nodes be called leaves. Initially, information will have the stabilizer elements for the classical repetition code: ⊗k logical operators Z¯ = Zr and X¯ = XrZr0 . The operation b b . When we turn on the X terms on the non- of the amplifier is as follows. First, single-qubit X terms |leavesi → | wei will thus have taken the information at the are turned on while turning off all of the graph stabiliz- root (and its child) and encoded it into this code (note P ¯ ¯ ers. Then a i Zi term is turned on across all of the that the roles of X and Z are reversed in this process leaves. One then waits for observed decay events from because of an extra Hadamard at the beginning of this the leaves: no decays indicates the system was in the 1 process). Stabilizer code arguments [15, 16] make this eigenstate of X¯ initially; O(#leaves) decays indicates the− statement rigorous: we promote the graph stabilizer to system was in the +1 eigenstate of X¯ initially. The util- logical Zs and then the local X terms become products ity of this device is that it turns information encoded into of logical Xs. The information as described by the en- one qubit into information copied (in a particular basis) coding above is untouched by this process and thus the to many qubits, thus facilitating the measurement of this adiabatic evolution described above does not affect this information. encoded information. To see how this process works, we note that, while we An equivalent way to derive this graph is to take defined the information as initially encoded into Z¯ = Zr the circuit for copying information recursively using and X¯ = XrZr0 , we could have chosen equivalent oper- controlled-NOTs and prepared ancilla states, convert it ators by multiplying by graph stabilizers. Thus, if we to a graph state, and then simplify this state by noting, 14 for instance, that two nodes on a line can be eliminated 2 from a graph because they correspond to H = I, where A 1 H is the Hadamard gate. 2 o 1' B. Adiabatic Router B It is convenient to design a device that can route quan- tum information in making a modular clocked architec- ture. The goal of the router construction is to design a o o system such that application of a field across different portions of the device can be used to steer the quantum information conditional on where the field has been ap- plied. The basic method for achieving this goal is the gadget shown in Fig. 10. The gadget shown can be used to produce a state, conditional on where in the device a C field is applied. Initially the Hamiltonian for this gadget is a sum over graph stabilizers for all vertices: 1 1' 2 2' H = (Z2X1 Z2X10 + Z2X + Z1Z10 Z X2) (25) i − − o o Then this Hamiltonian is adiabatically turned off while either of the two following Hamiltonians is turned on: FIG. 10. Gadgets and Router. Here we show the stages in understanding a router. In (A) we show the basic gadget H = (X1 + X2) (26) f − that can be used to conditionally prepare a ±1 eigenstate of Xo, depending on where the field is applied. (B) is the or same construction, but now demonstrating the flexibility of 0 the construction to the applied field. The applied field need H = (X10 + X2) . (27) f − only cover the desired leg in order to get the correct condi- tional behavior. (C) is a non-optimized router construction. The point of this process is that, depending on which If we apply the field everywhere but one of the two circled of these is turned on, the state of the qubit o will differ regions, this will route the quantum information incoming at between being a +1 eigenstate of X and a 1 eigenstate 0 0 − points 1 and 2 and output it at points 1 and 2 , applying of X. either the identity gate (1, 2) → (10, 20) or the swap gate, To see this result, first consider the case of ending in (1, 2) → (20, 10), depending on which circled region contains π Hf . Define a stabilizer code using the following operators a nonzero field. Here the black nodes are twisted by 4 , the π grey nodes by − 4 , and the gradient-shaded node by π. S1 = X1X10 ,S2 = X1Xo,

Z¯1 = Z1Z10 ZoX2, X¯1 = X1, system will be in the +1 eigenvalue eigenspaces of S , Z¯2 = Z2X1, X¯2 = X2 (28) 2 X¯1, and X¯2, while being in the 1 eigenvalue eigenspace ¯ − where S1 and S2 are the stabilizers for the code and Pj of S1. In particular, Xo = X¯1S2, which implies that the are the encoded Pauli operators for this code. Then we system is in the eigenstate of Xo with eigenvalue +1. 0 see that we can express the initial Hamiltonian as On the contrary, consider the case ending in Hf . De- fine the code similarly to above, Hi = (Z¯1 + Z¯2(I S1 + S2)) (29) − − 0 0 S1 = X1X10 ,S2 = X10 Xo, and the final Hamiltonian as ¯0 ¯ 0 Z1 = Z1Z10 ZoX2, X1 = X10 , 0 H = (X¯1 + X¯2) (30) Z¯ = Z2X10 , X¯2 = X2 . (31) f − 2 Since S1 and S2 commute with these Hamiltonians, the Then we can express the initial Hamiltonian as subspaces defined by the eigenvalues of these operators H = Z¯0 + Z¯0 ( I + S0 + S0 )
(32) will remain constant. The initial Hamiltonian will have a i − 1 2 − 1 2 ground state that is in the 1 eigenstate of S1 and the +1 − and the final, primed, Hamiltonian as eigenstate of S2. Initially the system will also be in the +1 eigenvalue eigenspace of Z¯1 and Z¯2. Upon applica- 0 ¯ 0 ¯ 0
Hf = X1 + X2 . (33) tion of the fields as represented by Hf , the information − in these later two encoded qubits will be adiabatically From these expressions one can deduce that the system ¯0 0 0 dragged (with no energy level crossings) to +1 eigenval- will start in the 1 eigenstate of the Z2, S1, and S2 ¯ ¯ − ¯0 ues of X1 and X2. Thus at the end of this evolution the operators and in the +1 eigenstate of Z1. At the end of 15 the evolution the system will end up in the +1 eigenstate VI. CONCLUSION ¯ 0 ¯ 0 0 0 of X1 and X2 and in the 1 eigenstate of S1 and S2. Since ¯ 0 0 − Xo = X1S2, this implies that at the end of this adiabatic In conclusion, we have introduced a new method for evolution the system will be in the 1 eigenstate of X . − o building a quantum computer based upon the notion of an adiabatic quantum transistor. Two notable benefits Thus we see that depending on whether the final of this method are that the system is robust to timing er- Hamiltonian is H or H0 , the qubit located at o is in f f rors (as in universal adiabatic QC) and that it is modular either the +1 or 1 eigenstate of X . Note that this o in nature (something that prior universal adiabatic QC depends only on where− the field H is applied. Further- f models lacked). Instead of requiring increasingly accu- more, note that there is flexibility in spatially achieving rate timing and control mechanisms, this model requires this result. For example, in Fig. 10(B) we show a larger one to focus on increasing the fabrication quality of en- version of this gadget. Depending on whether the ap- gineered interactions in many-qubit systems. We have plied field is in either of the two circles, this produces argued that the noise model for our scheme will follow a +1 or 1 eigenstate of the last qubit at o. The ex- an independent error model and thus is amenable to sta- act location− of this applied field is not important, except bilization by standard methods of fault-tolerant quantum for the point that the field entirely spans one of the two computing. While our actual construct is not optimized “legs” in the construction. Of course, while we would for current experimental implementation, the mere exis- ideally have a field profile that exactly vanishes outside tence of devices like the one we describe, combined with the oval in Fig. 10, imperfections in the field will affect recent experimental progress in building highly control- the qubits on the boundary; we leave open the question lable quantum simulators [45], gives us hope that adia- of quantifying the errors introduced by realistic control batic quantum transistors are a viable new path toward fields. building a large-scale quantum computer. One can extend this idea to then create a router where the routing depends only upon where a field has been applied. The basic idea is rather simple: if one has the ACKNOWLEDGMENTS ability to conditionally create one of the two orthogonal states by the location where the field is applied, then DB was supported by the NSF under grants 0803478, one can use this as input into a controlled swap gate. 0829937, and 0916400. DB and GMC were supported In Fig. 10(C) we show, for example, such a construction by DARPA under QuEST grant FA-9550-09-1-0044. (no attempt has been made to optimize this construc- STF was supported by the Australian Research Council tion) based upon the conditional swap gate construction Centre of Excellence for Engineered Quantum Systems of Smolin and DiVincenzo [44]. If one applies a field ev- CE110001013 and by the Office of the Director of Na- erywhere but in one of the two circled regions, then this tional Intelligence and Intelligence Advanced Research routes the quantum information depending on which of Projects Activity (IARPA) through the Army Research the two circled regions is left out. In particular, infor- Office. GMC was supported by a Computational Sci- mation coming into 1 and 2 is thus routed (permuted) to ence Graduate Fellowship under DoE grant DE-FG02- output information at 10 and 20. 97ER25308.

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It is convenient to add an extra qubit, called the 0th qubit, 17 and consider the Hamiltonian Note the second Hamiltonian follows by direct calcula- tion, but also because of the traceless nature of H0(s). l  l−1  l 0 X X Explicitly, we find that H (s) = (1 s) [Z] s [X] [X] +1 (A2) l − − i − i i i=1 i=0 2 2 M = s δ (1 δ ) + (1 s) δ (1 δ 0) ij i,j − i,l − i,j − i, Then, when we restrict to the +1 eigenspace of [X]0 + s(1 s)(δ +1 + δ +1) . (A10) − i,j j,i ([X]0 commutes with the Hamiltonian) we will obtain Ql It is easy to check that the (unnormalized) vector Hl(s). Also note that if we conjugate Hl(s) by i=1[Z]i, we will obtain Hl(s) with the [X]1 term flipped in sign. l l−i X s 1 Therefore the 1 eigenvalue of the [X]0 eigenspace has Φ = − i (A11) − | i s | i the exact same spectrum as the +1 eigenvalue of the [X]0 i=0 eigenspace. Thus we know that H0(s) will have exactly l is an eigenvector of M with eigenvalue 0 for all values the same spectrum as Hl(s) but will be two-fold degener- ate, with the degeneracy corresponding to the eigenvalue of s. This implies that there is always a mode with zero energy, which is exactly a consequence of the fact that the of [X]0. Thus, if we are interested in the gap of Hl(s) we 0 Hamiltonian is always two-fold degenerate. As argued can work equally well with Hl (s), which we now assume. 0 above, we can ignore this fact. The computation of the energy spectrum of Hl (s) fol- lows the techniques of Lieb, Schultz, and Mattis [25]. We can find the eigenvalues of M by using an ansatz After a Jordan-Wigner transform [46], the Hamiltonian of the form can be written as l X h 1
i l−1 v = sin θ i + + φ i . (A12) X | i 2 | i H0(s) = s c† c
c† + c
i=0 l − i − i i+1 i+1 i=0 If we apply M to this vector then we obtain three equa- l X tions: two boundary terms and a term from the bulk. (1 s) 2c†c I
(A3) − − i i − Define γ = s(1 s). The bulk term gives us the equation i=1 − h i h i γ sin θi 1
+ φ + (1 2γ λ) sin θi + 1
+ φ We can express this as − 2 − − 2 h i l + γ sin θi + 3
+ φ = 0 , (A13) X h 1 i 2 H0(s) = A c†c + B c†c† + B c c
+ ΓI l i,j i j 2 i,j i j j,i i j i,j=0 where λ is the eigenvalue. After some tedious math, this (A4) can be turned into where A is an (l + 1) (l + 1) symmetric matrix given by λ = (1 2γ) + 2γ cos θ . (A14) × − A = s(δ +1 + δ +1) 2(1 s)δ (1 δ 0), (A5) The two boundary terms give the equations ij − j,i i,j − − i,j − i, B a (l + 1) (l + 1) antisymmetric matrix given by s2 sin θ + φ
+ γ sin 3θ + φ
= λ sin θ + φ
(A15) × 2 2 2 B = s(δ +1 δ +1), (A6) and ij − j,i − i,j h 1
i 2 h 1
i and Γ = (1 s)l. Then we can find new fermion oper- γ sin θ l 2 + φ + (1 s) sin θ l + 2 + φ − † − − ators, ηk which are linear combinations of the c and c i i h 1
i such that = λ sin θ l + 2 + φ . (A16)

l l l ! X 1 X 1 X This first equation can be rearranged to yield H0(s) = ω η†η + A ω + Γ I l k k k 2 ii − 2 k k=0 i=0 k=0 θ
θ
(1 s) sin 2 + φ = s sin 2 φ (A17) (A7) − − while the second one can be manipulated to become or, after simplifying, h 3
i h 1
i l (1 s) sin θ l + 2 + φ = s sin θ l + 2 + φ . X  1 − − H0(s) = ω η†η (A8) (A18) l k k k − 2 k=0 1−s Solving these equations for s we obtain the equation with the ωks being the square roots of the eigenvalues of h i 4M [25], where θ
1
sin φ sin θ l + 2 + φ 2 − = − . (A19) 1 θ
h 3
i M = (A + B)(A B) . (A9) sin 2 + φ sin θ l + + φ 4 − 2 18

This can be reduced to Thus we see that the eigenvalues are

cosθ(l + 2) = cos(θl), (A20) 2  h kπ
i ωk(s) = 4 1 2s(1 s) 1 cos l+1 (A22) which has solutions for − − − πk θ = (A21) for k = 1, 2, . . . , l. Each of these is minimized for s = 1/2, l + 1 for which the eigenvalues become where k is an integer. When k = 0, this equation does not have a solution for general s. Fortunately, we already 2 kπ
2 kπ
ωk(1/2) = 2 + 2 cos +1 = 4 cos 2( +1) . (A23) have the k = 0 eigenvector, so we have found the rele- l l vant nonzero eigenvalues. For k = 0, the phase shift φ 6 can be found by solving the transcendental equation of From this we see that ωk’s smallest value is O(1/l), oc- Eq. (A17). curring when k = l.