A Fault-Tolerant Scheme of Holonomic Quantum Computation on Stabilizer Codes with Robustness to Low-weight Thermal Noise
AA Fault-Tolerant Scheme of Holonomic Quantum Computation on Stabilizer Codes withRobustness to Low-weight Thermal Noise
Yi-Cong Zheng ∗ and Todd A. Brun † Department of Electrical Engineering,Center for Quantum Information Science & Technology,University of Southern California,Los Angeles, California, 90089 (Dated: September 24, 2018)We show an equivalence relation between fault-tolerant circuits for a stabilizer code and fault-tolerant adia-batic processes for holonomic quantum computation (HQC), in the case where quantum information is encodedin the degenerated ground space of the system Hamiltonian. By this equivalence, we can systematically con-struct a fault-tolerant HQC scheme, which can geometrically implement a universal set of encoded quantumgates by adiabatically deforming the system Hamiltonian. During this process, quantum information is pro-tected from low weight thermal excitations by an energy gap that does not change with the problem size.
PACS numbers: 03.65.Vf, 03.67.Lx, 03.67.Pp
I. INTRODUCTION
Quantum computers are superior to classical ones in solv-ing specific difficult problems, yet they are extremely vulner-able to errors during the computation process. It has beenshown that if the errors of each type are local, and their ratesare below a certain threshold, it is possible to implement quan-tum algorithms reliably with arbitrarily small error [1–4].These quantum threshold theorems are based on the idea ofquantum error correction (QEC).In addition to QEC, there have also been proposals to dealwith noise by designing the “hardware” to provide inherent ro-bustness. One of such method is holonomic quantum comput-ing (HQC) [5]—an all-geometric, adiabatic method of com-putation that uses a non-Abelian generalization of the Berryphase [6]. This approach is robust against certain types oferrors during the adiabatic evolution [7–9] and offers somebuilt-in resilience.Another method is to use the adiabatic quantum computing(AQC) [10, 11] model instead of the standard quantum circuitmodel, which slowly drags the ground state of the system tothe final Hamiltonian, whose ground state encodes the solu-tion of the problem. AQC would take advantage of the energygap between the ground state and other excited states to sup-press thermal noise when evolution is very slow [12, 13].The combination of fault-tolerant techniques and HQC wasstudied in Ref. [14, 15], where the system Hamiltonian is anelement of the stabilizer group or gauge group. Single qubitor two-qubit unitary operations are realized through continu-ously deforming the the system Hamiltonian. During this pro-cess, the path in the parameter space forms an open loop andresults in the desired unitary transformation. After a sequence ∗ Electronic address: [email protected] † Electronic address: [email protected] of such elementary operations, a closed-loop holonomy is ob-tained in the code space. However, this approach does notprotect quantum information from thermal noise since there isno energy gap between the code space and error spaces. Also,while considerable work has been done in [12, 16], a fault-tolerant theory for AQC is still lacking. The system’s mini-mal energy gap, which determines the time scale of evolution,scales as an inverse polynomial in the problem size [17, 18],so that the temperature must be lowered polynomially to pre-vent thermal excitation.In this paper, we present a scheme combining advantagesof all three methods mentioned previously. First, we show anequivalence relation between fault-tolerant circuits and fault-tolerant adiabatic processes in the case where quantum infor-mation is encoded in a code space, which is also the groundspace of a system Hamiltonian. Based on this, we presentan alternative way to systematically construct a fault-tolerantHQC process that takes advantage of the energy gap betweenthe ground space and other excitation states. Unlike AQC,this gap does not change with the problem size, and we knowthe exact value of the gap during the process, which greatlyenhances the ability to prevent low-weight thermal excita-tion. With a lower error rate at the physical level of the fault-tolerant scheme, it may help to reduce the number of qubitsneeded and the frequency of error detection and error correc-tion.The structure of this paper is as follows. In Sec. II, wereview the preliminaries that we will use to formulate ourproblem. Specifically, after defining HQC in Sec. II A, wereview the geometrical setting of the holonomic problem inSec. II B, and the basic ideas of fault-tolerant quantum com-puting in Sec. II C. We connect fault-tolerant techniques andHQC in Sec. III. In Sec. III A, we describe our method to con-struct an adiabatic process from a fault-tolerant circuit to im-plement encoded unitary operations. Then in Sec. III B, weprove that our method of constructing encoded unitary op-erations is fault-tolerant, and discuss how it can realize uni- a r X i v : . [ qu a n t - ph ] J a n versal fault-tolerant quantum computation. Several examplesare given in Sec. IV. In Sec. IV A, we show how our schemeworks on the simplest 3-qubit repetition code. A less trivialexample, of the encoded CNOT gate for the Steane code, isgiven in Sec. IV B. We summarize our results and conclude inSec. V. II. PRELIMINARIESA. Holonomic quantum computation
Consider a family of Hamiltonians { H λ } on an N − dimensional Hilbert space. The point λ , parametriz-ing the Hamiltonian, is an element of a manifold M calledthe control manifold, and the local coordinates of λ aredenoted by λ i ( ≤ i ≤ dim M) . Assume there are onlya fixed number of eigenvalues ε k ( λ ) (this is the case weare interested in) and suppose the n th eigenvalue ε n ( λ ) is K n -fold degenerate for any λ . The degenerate subspace at λ is denoted by H n ( λ ) . The orthonormal basis vectors of H n ( λ ) are denoted by {∣ φ nα ; λ ⟩} , satisfying H λ ∣ φ nα ; λ ⟩ = ε n ( λ )∣ φ nα ; λ ⟩ , (1)and ⟨ φ nα ; λ ∣ φ mβ ; λ ⟩ = δ nm δ αβ . (2)Now assume the parameter λ is changed adiabatically, whichmeans that ( ε n ( λ ( t )) − ε n ′ ( λ ( t ))) T ≫ (3)is satisfied for n ≠ n ′ during ≤ t ≤ T ). Suppose the ini-tial state at t = is an eigenstate ∣ ψ n ( )⟩ = ∣ φ nα ; λ ( )⟩ . TheSchr¨odinger equation is i dd t ∣ ψ n ( t )⟩ = H ( λ ( t ))∣ ψ n ( t )⟩ , (4)whose solution will have the form ∣ ψ n ( t )⟩ = K n ∑ β = ∣ φ nβ ; λ ( t )⟩ U βα ( t ) . (5)where we have used the adiabatic approximation from Eq. (3).Substituting Eq. (5) into Eq. (4), one finds that U βα satisfies ˙ U βα ( t ) = − iε n ( λ ( t )) U βα ( t )− ∑ µ ⟨ φ nβ ; λ ( t )∣ dd t ∣ φ nµ ; λ ( t )⟩ U µα ( t ) . (6)The solution can be expressed as U ( t ) = exp (− i ∫ t ε n ( λ ( s )) d s ) T exp (− ∫ t A n ( τ ) d τ ) , (7) where T is the time-ordering operator and A nβα ( t ) = ⟨ φ nβ ; λ ( t )∣ dd t ∣ φ nα ; λ ( t )⟩ (8)is the Wilczek-Zee (WZ) connection [6]. Define the connec-tion A ni,βα ( t ) = ⟨ φ nβ ; λ ( t )∣ ∂∂λ i ∣ φ nα ; λ ( t )⟩ , (9)through which U ( t ) can be expressed as U ( t ) = exp (− i ∫ t ε n ( λ ( s )) d s ) P exp (− ∫ λ ( t ) λ ( ) ∑ i A ni d λ i ) , (10)where P is the path-ordering operator. Suppose the path λ ( t ) is a loop λ in M such that λ ( ) = λ ( T ) = λ . Then aftertransporting through λ , states are transformed to ∣ ψ n ( T )⟩ = K n ∑ β = ∣ ψ nβ ( )⟩ U βα ( T ) . (11)The unitary matrix Γ λ = P exp (− ∮ λ ∑ i A ni d λ i ) (12)is called the holonomy associated with the loop λ ( t ) . Γ λ is a purely geometric object, and is independent of theparametrization of the path. Note that for a given Γ λ , thereexist infinitely many paths λ . Given a path λ , to find theholonomy is easy. However, the inverse problem—given aholonomy, to find the the proper path λ —is in general nottrivial at all. In the rest of the paper, we will discuss how tofind a proper path λ to realize a certain holonomy in the codespace, and thus perform an encoded quantum gate operation. B. Formulation of geometric problem
The definition introduced in Sec. II A is not easy to use forour purpose. In this section, we outline the geometric set-ting of the holonomic problem as described in Refs. [19, 20],which gives a clearer picture and more concise formulation ofthe problem. We focus on the ground space of the Hamilto-nian to simplify the problem. However, this formalism is quitegeneral, and can be applied to any eigenspace of the systemHamiltonian.Suppose we have a family of Hamiltonians acting on theHilbert space C N , and the ground state of each Hamilto-nian is K -fold degenerate ( K < N ). The natural math-ematical setting to describe this system is the principalbundle ( S N,K ( C ) , G N,K ( C ) , π, U ( K )) , which consists ofthe Stiefel manifold S N,K ( C ) , the Grassmann manifold G N,K ( C ) , the projection map π ∶ S N,K ( C ) → G N,K ( C ) ,and the unitary structure group U ( K ) . We explain the mean-ing of these mathematical objects below.The Stiefel manifold is defined as: S N,K ( C ) = { V ∈ M ( N, K ; C )∣ V † V = I K } , (13)where M ( N, K ; C ) is the set of N × K complex matricesand I K is the K − dimensional unit matrix. Physically, eachcolumn of V ∈ S N,K ( C ) can be viewed as a normalized statein C N , and V can be viewed as an orthonormal set of K basisof the ground space of Hamiltonian. Since we have freedom totransfer from one orthnormal basis of ground space to anotherthrough unitary transformation, we can then define the unitarygroup U ( K ) that acts on S N,K ( C ) from the right: S N,K ( C ) × U ( K ) → S N,K ( C ) , ( V, h ) ↦
V h, (14)by the matrix product of V and h . V and V h can be viewed astwo different orthonormal basis corresponding to the groundspace.During the adiabatic evolution, the ground space of theHamiltonian may change. The ground space can be viewedas a K -dimensional hyperplane in C N . So we introduce theGrassmann manifold in C N : G N,K ( C ) = { P ∈ M ( N, N ; C )∣ P = P, P † = P, Tr P = K } , (15)where P is a projection operator onto the hyperplane in C N ,and the condition Tr P = K indicates that the hyperplane is K -dimensional. In our scenario, P ∈ G N,K ( C ) can be regardedas the projector onto the K -dimensional ground space of theHamiltonian.The relationship of the orthonormal basis V and groundspace P can be seen as follows. We define the projection map π ∶ S N,K ( C ) → G N,K ( C ) as π ∶ V ↦ P ∶= V V † . (16)So, given an orthonormal basis, we can obtain the correspond-ing ground space projector. We can check that the basis V andbasis V h with h ∈ U ( K ) belong to the same ground space,since π ( V h ) = (
V h )( V h ) † = V hh † V † = V V † = π ( V ) . (17)In our scenario of HQC, we want to transform the groundspace adiabatically during the procedure. To formulate sucha process, we also define the left action of the unitary group U ( N ) on both S N,K ( C ) and G N,K ( C ) by the matrix prod-uct: U ( N ) × S N,K ( C ) → S N,K ( C ) , ( g, V ) ↦ gV, (18)and U ( N ) × G N,K ( C ) → G N,K ( C ) , ( g, P ) ↦ gP g † . (19)It is easy to see that π ( gV ) = gπ ( V ) g † . This action is transi-tive: there is a g ∈ U ( N ) for any V, V ′ ∈ S N,K ( C ) such that V ′ = gV . There is also a g ∈ U ( N ) for any P, P ′ ∈ G N,K ( C ) such that P ′ = gP g † . So this action is sufficient to describeany ground space transformation.The canonical connection form on S N,K ( C ) is defined as a u ( K ) -valued one-form on G N,K ( C ) : A = V ( P ) † d V ( P ) , (20) which is a generalization of the WZ connection in Eq. (8).This is the unique connection that is invariant under the trans-formation in Eq. (14): ˜ A = h † V ( P ) † d ( V ( P ) h )= h † Ah + h † d h, (21)which can be recognized as a gauge transformation.Now we apply this formalism to the system dynamic ofHQC. The state vector ψ ( t ) ∈ C N evolves according to theSchr¨odinger equation, i dd t ψ ( t ) = H ( t ) ψ ( t ) . (22)The Hamiltonian has a spectral decomposition, H ( t ) = L ∑ l = ε l ( t ) P l ( t ) , (23)with projection operators P l ( t ) . Therefore, the set of en-ergy eigenvalues ( ε ( t ) , . . . , ε L ( t )) and orthogonal projec-tors ( P ( t ) , . . . , P l ( t )) encodes the information of the controlparameters of the system. For the ground space, without lossof generality, the energy is assumed to be zero: ε ( t ) = .We write P ( t ) as P ( t ) for simplicity. Suppose the de-generacy K = Tr P ( t ) is constant. For each t , there exists V ( t ) ∈ S N,K ( C ) such that P ( t ) = V ( t ) V † ( t ) . By the adi-abatic approximation, we can substitute for ψ ( t ) ∈ C N a re-duced state vector φ ( t ) ∈ C K : ψ ( t ) = V ( t ) φ ( t ) . (24)Since H ( t ) ψ ( t ) = ε ( t ) ψ ( t ) = , the Schr¨odinger equation(22) becomes d φ d t + V † d V d t φ ( t ) = , (25)and the solution can be represented formally as φ ( t ) = P exp (− ∫ V † d V ) φ ( ) . (26)Therefore, ψ ( t ) can be written ψ ( t ) = V ( t )P exp (− ∫ V † d V ) V † ( ) ψ ( ) . (27)In particular, if the system comes back to its initial point, as P ( T ) = P ( ) , the holonomy Γ ∈ U ( K ) is defined as Γ = V † ( ) V ( T )P exp (− ∫ V † d V ) , (28)and the final state is ψ ( T ) = V ( ) Γ φ ( ) . (29)According to the formula above, an operation Γ ∈ U ( K ) isapplied to the ground space.If the condition V † ⋅ d V d t = (30)is satisfied for all t , the curve V ( t ) in S N,K ( C ) is called ahorizontal lift of the curve P ( t ) = π ( V ( t )) in G N,K ( C ) .Thenthe holonomy (28) is greatly simplified to Γ = V † ( ) ⋅ V ( T ) ∈ U ( K ) . (31)Now we are ready to reformulate the problem stated atthe end of Sec. II A. Given a desired unitary operation U op ∈ U ( K ) and a fixed initial point P ( ) ∈ G N,K ( C ) , we want tofind a loop P ( t ) ∈ G N,K ( C ) with base points P ( ) = P ( T ) whose horizontal lift V ( t ) ∈ S N,K ( C ) produces the holon-omy Γ = U op according to Eq. (31). In Sec. III A, we willdiscuss in detail how to find such a loop P ( t ) whose horizon-tal lift gives the desired holonomy in the code space. { holonomy horizonal lift general curve G N,k
FIG. 1: Horizontal lift as a specified curve in S N,K ( C ) whose pro-jection is P ( t ) . The initial condition V ( ) becomes V ( T ) , which isgenerally different from V ( ) . The difference is the holonomy. A visualization of a horizontal lift is shown in Fig. 1. With-out loss of generality, we can always restrict ourselves to thecase where P ( t ) has the form P ( t ) = U ( t, ) P ( ) U † ( t, ) = U ( t, ) v v † U † ( t, ) , (32)for some smooth U ( t, ) ∈ U ( N ) according to Eq. (19). Notethat, U ( t, ) should be chosen such that, at any time tU ( t + τ, t ) P ( t ) U † ( t + τ, t ) ≠ P ( t ) , (33)for some neighborhood of t . This condition can also be statedas [ ∂∂τ U ( t + τ, t )∣ τ = , P ( t )] ≠ . (34) The case where Eq. (34) equals 0 is allowed only at a finitenumber of points in [ , T ] . The horizontal curve should sat-isfy the following set of equations: V † ⋅ d V d t = ,P ( t ) = V ( t ) V † ( t ) = U ( t, ) v v † U † ( t, ) . (35)The general solution to these equations can be written as: V ( t ) = U ( t, ) v h ( t, ) (36)for some h ( t, ) ∈ U ( K ) . Substituting Eq. (36) into Eq. (35)we get: ˙ h ( t, ) = − v † U † ( t, ) ˙ U ( t, ) v h ( t, ) . (37)A well known result of differential geometry about theuniqueness of a horizontal lift curve [21] can now be directlyproved in this specific scenario, which will be used later. Lemma 1.
Let P ∶ [ , T ] → G N,K ( C ) be a curve in G N,K ( C ) and let v ∈ π − ( P ( )) . Then there exists a uniquehorizontal lift V ( t ) in S N,K ( C ) such that V ( ) = v .Proof. It’s easy to show that v † U − ( t, ) ˙ U ( t, ) v is anti-Hermitian, so h ( t, ) ∈ U ( K ) for all t . Define V ′ ( t ) = U ( t, ) V ( ) to be a particular curve in principal bundle thatgives a corresponding WZ connection A ′ = V ′ † d V ′ . With ini-tial condition h ( , ) = I K , the solution of Eq. (37) can bewritten as: h ( t, ) = P exp (− ∫ A ′ ) , (38)and hence there exists an unique horizontal lift V ( t ) . C. Stabilizer codes and fault-tolerant computation
A quantum error-correcting code is formally defined as asubspace C of some larger Hilbert space. A necessary andsufficient condition for a set of errors { E i } to be correctableis [1, 4]: P E † i E j P = α ij P, ∀ i, j, (39)for some Hermitian matrix α . Here P is the projector onto C .Since any linear combination of { E i } is also correctable, wedefine E =
Span { E i } (40)to be a correctable error set for code C .The codes we are interested in are the stabilizer codes [22].We briefly review the formalism of stabilizer codes. Let G n be the Pauli group acting on n qubits. An Abelian subgroup S of G n is called a stabilizer group if − I ∉ S . The stabilizergroup defines a subspace of the n -qubit Hilbert space by C = {∣ ψ ⟩ ∶ S ∣ ψ ⟩ = ∣ ψ ⟩ for all S ∈ S} . (41)This C is called the code space. C is nonzero since − I ∉ S . Astate in C is called a codeword. This subspace is the simulta-neous +1 eigenspace of the stabilizer group. If the subspacehas dimension k ( k logical qubits), the stabilizer code canbe specified by n − k commuting stabilizer generators, whichare elements of G n . The group S can be represented by thesestabilizer generators: S = ⟨{ S j }⟩ . All stabilizer codes canbe characterized by three parameters [[ n, k, d ]] , where d isthe minimum distance of the code, which is equal to the mini-mum weight of all nontrivial elements in the normalizer groupof S .With the use of stabilizer codes, it is possible to build aquantum processor that is fault-tolerant [1, 4, 22]. A quan-tum information processor is called fault-tolerant if the infor-mation is encoded in a quantum error-correcting code at alltimes during the procedure, and a failure at any point in theprocedure can only propagate to a small number of qubits,so that error correction can remove the errors. It has beenshown that fault-tolerant computation is possible on any sta-bilizer code [4, 22] for some error model. Typically, there arethree elementary quantum “gadgets”: encoded state prepara-tion, encoded unitary operations and encoded state measure-ment. Through enlarging or concatenating the fault-tolerantgadgets, a computation can achieve arbitrary accuracy, if theerror rate is low enough [4]. Encoded Clifford unitary oper-ations play a key role in fault-tolerant computation, since formost proposed schemes of fault-tolerant quantum computa-tion, like concatenation of the Steane code [1], C4 code [23]or surface code [24], we can prepare encoded non-Cliffordmagic states using techniques like state distillation, which canbe implemented by encoded Clifford unitary operations. Sowe will focus on encoded Clifford operations and their holo-nomic implementation. . . . . . .g g g p - 1 g p FIG. 2: A logical unitary quantum operation is realized as a seriesof quantum gates from a universal set of gates in the circuit model.
In the standard circuit model, an encoded unitary oper-ation can be realized by a series of quantum gates, say p gates chosen from a universal set of gates as shownin Fig. 2. Commonly, the universal set of gates U ={ Hadamard , CNOT , S, π / } is used to describe the circuit,which is good for certain fault-tolerant schemes. Here, wechoose another universal set: U = { R x = exp (− i π X ) , R zz = exp ( i π Z ⊗ Z ) , S, π / } , (42)which proves to be much more suitable for our adiabaticscheme. Errors can occur anytime during the process, both between and during gate operations. Noisy gates are alwaysequivalent to a perfect gate followed by an error operator, sowe can just focus on the errors occurring between the gates.If an error E q ∈ E occurs between gates q − and q , it willpropagate to E q ′ = p ∏ l = q g p + q − l ⋅ E q ⋅ ⎛⎝ p ∏ l = q g p + q − l ⎞⎠ † . (43)If a circuit is fault-tolerant, we can suppose that E q ′ would bestill in the same correctable error set E . According to this ob-servation, we give a generalized definition of a fault-tolerantcircuit for a code C : Definition 1.
Given a code C and a particular correctableerror set E for this code, a circuit G that realizes an encodedunitary operation is called a fault-tolerant circuit for C if forany ≤ q < p , U qp = ∏ pl = q g p + q − l maps any subset of E toanother subset of E . This definition of a fault-tolerant circuit may be, however,too strong. In practice, it may be very difficult to find such acode and corresponding circuit. For a practical error model,strongly correlated errors happen with much lower probabilitythan weakly correlated or local ones, so we will focus on localerrors. For example, if our codes are stabilizer codes, the errorset E local can be spanned by Pauli operators with weight lessthan ⌊ d − ⌋ , which occur with relatively high probability. Ifwe limit ourselves to such a high-probability correctable localerror set E local , then we get a weaker version of the definitionof a fault-tolerant circuit: Definition 2.
Given a stabilizer code [[n,k,d]] with a cor-rectable error set E , and there is a high-probability local er-ror set E local ⊂ E , a circuit G that realizes an encoded unitaryoperation is called a fault-tolerant circuit for this code if forany ≤ l < p , U qp = ∏ pl = q g p + q − l maps E local to some subset of E . Remark 1.
According to this definition, the encoded fault-tolerant unitary circuit does not necessarily need to betransversal, although the reverse is always true. If a circuitbuilt of gates from U is fault-tolerant, then we can decom-pose its gates into gates from U , and the new circuit weobtain is also fault-tolerant. So, in the rest of the paper, weassume that the given circuits are composed of gates from U . Remark 2.
We should mention here that in the followingdiscussion, we only consider circuits that contain no π / gates. In other words, we limit ourselves to Clifford circuits,since non-Clifford circuits will cause tremendous complexity.This restriction will be further discussed in Sec. III B. For-tunately, fault-tolerant encoded Clifford operations for stabi-lizer codes are made of Clifford circuits, that do not contain π / gates. Encoded non-Clifford operations usually do con-tain π / gates. We will not directly implement encoded non-Clifford operations, but instead make use of magic state dis-tillation, so this is not a serious restriction. III. FAULT-TOLERANT HOLONOMIC QUANTUMCOMPUTATION
To combine the advantages of holonomic quantum com-putation with fault-tolerant computation techniques, the basicidea is to obtain a holonomy on the code space, which is theground space of the system Hamiltonian, during an adiabaticevolution. One must make sure that the encoded quantuminformation is protected by a suitable error-correcting codethroughout the Hamiltonian deformation. For simplicity, weassume that error correction is applied at the end of the cyclicadiabatic evolution. However, this may not necessarily be truein practice. We require that an error occurring during the de-formation be correctable at the end:
Proposition 1.
Given a code C with a correctable error set E , suppose the initial state is ∣ ψ ( )⟩ ∈ C , and the deforma-tion of the Hamiltonian is adiabatic. Then, in general, eacheigenspace will undergo some transformation. Given a de-sired encoded operation (in our case, a holonomy) Ω g on thecode space, suppose a series of errors { E t i } occur at times t i during the evolution. Then { E t i } is correctable only if thefinal state ∣ ψ ( T )⟩ ∝ E f Ω g ∣ ψ ( )⟩ , for some E f ∈ E . In the case when { E t i } is empty, the statement is obvi-ous. The fault-tolerance of this process is well defined in thecase when { E t i } just has one element, say E t . Followingthe spirit of fault-tolerant quantum computation in the circuitmodel, we define fault-tolerance for an adiabatic process: Definition 3.
Given a code C , defined by the ground space ofan initial Hamiltonian with a correctable error set E , a de-sired encoded operation (holonomy) Ω g , and an initial state ∣ ψ ( )⟩ ∈ C , the corresponding cyclic adiabatic process iscalled fault-tolerant if any E t ∈ E local occurring at time t leadsto a final state E f Ω a ∣ ψ ( )⟩ for some E f ∈ E . Unlike AQC, we need to measure the stabilizer generatorsand do error correction after a single or multiple cycles of en-coded operations. At those points, we turn off the Hamil-tonian and apply a standard error correction procedure. Ifthis scheme is robust to low-weight thermal noise, and alsoevolves slowly enough that the adiabatic error is well be-low the threshold (which we will examine in some detail inSec. III B), then the frequency of error recovery operationscan be greatly reduced.We will show how to construct a fault-tolerant adiabaticprocess to do a holonomic encoded quantum unitary opera-tion starting from a fault-tolerant circuit, and we will provethat such a process is fault-tolerant by Def. 3.
A. Scheme
Given a stabilizer code with stabilizer group S for n qubits( n = N ), we set the system Hamiltonian at the very begin- ning to be H ( ) = − ∑ j S j . (44)Thus the code space is the ground space of the Hamiltonianwith dimension K = k . We deform the Hamiltonian as fol-lows: H ( t ) = ∑ j C j ( t ) S j ( t )= ∑ j C j ( t ) U ( t, ) S j U † ( t, ) , (45)with S j ( t ) = U ( t, ) S j U † ( t, ) , and [ S i ( t ) , S j ( t )] = for all i , j . C j ( t ) ∈ [− , ] is the weight of S j ( t ) which is assumedto be controllable. The { S j ( t )} can be viewed as a set ofgenerators of an Abelian group, such as a stabilizer group.The Hamiltonian also has a spectral decomposition H ( t ) = ∑ s ε s ( t ) P s ( t ) . (46)Here, the { P s ( t )} are projectors onto the simultaneouseigenspace of all the S j ( t ) , with eigenvalues: ε s ( t ) = ∑ j C j ( t ) s j , (47)where the labels s j = ± form a vector: s = { s , s . . . s n − k } . (48)When the Hamiltonian changes, as shown previously inEq. (19), the ground space will also evolve. This defines atime-dependent code C t . Let P ( t ) = U ( t, ) P ( ) U † ( t, ) be the projector onto the ground space of the Hamiltonian H ( t ) such that s j = for all j . We emphasize that U ( t, ) should be chosen such that [ ∂∂τ U ( t + τ, t )∣ τ = , P s ( t )] ≠ for all s , (49)except for a finite set points t , so that the deformation proce-dure is non-trivial for all eigenspaces.This method will work only if the adiabatic condition foreach eigenspace P s is satisfied, so that each eigenspace under-goes some non-trivial holonomy during the cyclic evolution,in case an error takes the system to P s during the process. Thestandard adiabatic condition [25] can be reformulated for theeigenspace { P s α } : ∥ P s α ( t ) ∂∂t H ( t ) P s β ( t ) ∥ K ( ε s α ( t ) − ε s β ( t )) ≈ , for any α ≠ β. (50)This must hold for all t ∈ [ , T ] , where ∥ ⋅ ∥ is the tracenorm (∥ A ∥ = Tr √ A † A ) . For Hamiltonians of the form inEq. (45), it is very likely that different P s ( t ) ’s share the sameeigenvalues so the adiabatic condition would not be directlysatisfied. We will show later a systematic way to engineer thedeformation procedure so that each eigenspace P s ( t ) satisfiesthis condition during the adiabatic process.In addition, each eigenspace should undergo the sameholonomy to satisfy Prop. 1. Let’s see how it works. Define: U † ( t, ) ˙ U ( t, ) = iQ ( t, ) , (51)where Q ( t, ) is Hermitian. In order to obtain the same holon-omy for each P s , according to Eq. (37), P s ( ) Q ( t, ) P s ( ) should be related to P ( ) Q ( t, ) P ( ) in some way. If wecan make P s ( ) Q ( t, ) P s ( ) either equal to the zero matrixor proportional to P s ( ) for all s , then the character of thehorizontal lift of P s ( t ) is completely determined by U ( t, ) .Now we are ready to describe the scheme to construct afault-tolerant adiabatic process for a holonomic unitary oper-ation, starting from a fault-tolerant circuit. First, we divide thetime of evolution [ , T ] into p segments. The l th segment is [ t l − , t l ] , and we set t = and t p = T . Given a fault-tolerantcircuit G that realizes an encoded operation Ω g = ∏ pl = g p − l + ,we can follow the steps listed below:1. Set l = and t = .2. Check the number of S j ( t l − ) such that [ S j ( t l − ) , g l ] ≠ . If this number is odd, go to step 3, else go to step 4.3. For the l th time segment [ t l − , t l ] , choose a unitary op-erator U l ( t, t l − ) = g f l ( t ) l , with f l ∶ [ t l − , t l ] → [ , ] a monotonic smooth function with boundary conditions f ( t l − ) = and f ( t l ) = . We deform the Hamilto-nian such that H ( t ) = U l ( t, t l − ) H ( t l − ) U † l ( t, t l − ) inthe interval [ t l − , t l ] . All S j ( t l − ) are replaced at t l by S j ( t l ) = g l S j ( t l − ) g † l , and H ( t l ) = − ∑ j S j ( t l ) . Thengo to step 5.4. We need an additional operation to break the degen-eracy in this case, in order that the adiabatic condi-tion be satisfied for all P s ( t ) . From those S j ( t ) suchthat [ S j ( t l − ) , g l ] ≠ , we arbitrarily select one el-ement, say S b ( t l − ) , and change the Hamiltonian to H ( t ′ l − ) = H ( t l − ) + C b S b ( t l − ) . C b is a constantbetween 0 and 1; we will choose it to be 0.5. Thisprocedure can be done arbitrarily fast, and it will notaffect a state entirely contained in any P s , so we canjust set t ′ l − = t l − . We choose U l ( t, t l − ) = g f l ( t ) l inthis case, where f l ∶ [ t l − , t ′ l ] → [ , ] is a monotonicsmooth function with boundary conditions f l ( t l − ) = and f l ( t ′ l ) = . At time t ′ l , the Hamiltonian be-comes H ( t ′ l ) = − ∑ j S j ( t ′ l ) + C b S b ( t ′ l ) with S j ( t ′ l ) = g l S j ( t l − ) g † l . Then we remove the additional term inthe Hamiltonian, leaving H ( t l ) = − ∑ j S j ( t l ) , where S j ( t l ) = S j ( t ′ l ) . Again, this can be done arbitrarilyfast, so, we can set t ′ l = t l . Go to step 5.5. If l = p , the process is finished. Else, set l = l + and goto step 2.First, we will prove that in the case where no error happensduring the adiabatic evolution this process indeed gives an en-coded operation Ω g on the code space. For a circuit G , wedefine a set T (G) = { Z m } ⋃{ X m } ⋃{ Z m ⋅ Z m } , where m ranges over all qubits in G and m , m range over all pairs ofqubits shared by any two-qubit gates in G . Theorem 1.
Given a fault-tolerant circuit G defined for a sta-bilizer code C , with E ⊇ E local ⊃ T (G) , then following thesteps listed above we can perform a holonomic encoded oper-ation Ω g = ∏ pl = g p − l + for the code space P .Proof. It is easy to check that there is always a finite energygap between P ( t ) and any other P s ( t ) during the process.So if we choose the time scale properly, P ( t ) can alwayssatisfy the adiabatic condition. Consider the q th step of theimplementation. If the q th gate is single qubit gate, it acts onsome qubit m . If it is a two qubit gate, it acts on a pair ofqubits m and m . According to Eq. (51), for the q th step, wedefine i ˜ Q ( t, t q − ) = P ( t q − ) U † q ( t, t q − ) ˙ U q ( t, t q − ) P ( t q − ) . (52)Assume we are in step 3 (the argument for step 4 is the samewith a trivial modification). U q ( t, t q − ) is chosen to be g f q ( t ) q ,where g q is one of the gates from U . U q ( t, t q ) can be repre-sented explicitly for this set of gates as follows: U R xm q ( t, t q − ) = exp (− i π f q ( t ) X m ) ,U Z m Z m q ( t, t q − ) = exp ( i π f q ( t ) Z m Z m ) ,U S m q ( t, t q − ) = exp (− i π f q ( t ) Z m ) ,U π m q ( t, t q − ) = exp (− i π f q ( t ) Z m ) . (53)Define the code C q − as the ground space of H ( t q − ) , i.e., thespace projected onto by P ( t q − ) , and assume V ( t q − ) to bethe horizontal lift of P ( t q − ) = V ( t q − ) V † ( t q − ) . Accord-ing to Def. 1, ∏ pl = q g p + q − l maps E local to a subset of E , whichis the correctable error set of our code C , so it’s easy to checkthat E local is a correctable error set for code C q − , defined by P ( t q − ) . Since T (G) ⊂ E local , according to Eq. (39), wecould have: ˜ Q R xm ( t, t q − ) =− P ( t q − ) π f q ( t ) X m P ( t q − ) = α ( t ) P ( t q − ) , ˜ Q Z m Z m ( t, t q − ) = P ( t q − ) π f q ( t ) Z m Z m P ( t q − ) = α ( t ) P ( t q − ) , ˜ Q S m ( t, t q − ) =− P ( t q − ) π f q ( t ) Z m P ( t q − ) = α ( t ) P ( t q − ) , ˜ Q π m ( t, t q − ) =− P ( t q − ) π f q ( t ) Z m P ( t q − ) = α ( t ) P ( t q − ) . (54)It is easy to see that α r ( t ) , r =
1, 2, 3, 4 are all real.It is necessary to check that, for each step, Eq. (49) is satis-fied. We have P ( t ) = U q ( t, t q − ) P ( t q − ) U † q ( t, t q − ) . (55)We will just show the case where an R x gate is applied at the q th stage; the calculations for other gates are just the same. [ ∂∂τ U X m q ( t + τ, t )∣ τ = , P ( t )]= − i π [ ˙ f q ( t ) X m , P ( t )]= − i π U X m q ( t, t q − ) [ ˙ f q ( t ) X m , P ( t q − )] U X m † q ( t, t q − ) . (56)We multiply P ( t q − ) by [ X m , P ( t q − )] and have P ( t q − ) [ X m , P ( t q − )]= P ( t q − ) ( X m P ( t q − ) − P ( t q − ) X m )= − P ( t q − ) ( α ( t ) ˙ f q ( t ) π I + X m ) ≠ . (57)So, [ X m , P ( t q − )] ≠ , if X m ∉ ⟨ S j ( t q − )⟩ . This is in-deed true in this case, since for a well-defined circuit, X m ∉⟨ S j ( t q − )⟩ . Otherwise, R xm would have no effect at the q thstage. Then we have [ ∂∂τ U X m q ( t + τ, t )∣ τ = , P ( t )] ≠ , (58)when ˙ f q ( t ) ≠ .For any ˜ Q r ( t, t q − ) , from Eq. (37) we get ∂∂t h ( t, t q − ) = − iV † ( t q − ) ˜ Q r ( t, t q − ) V ( t q − ) h ( t, t q − )= − iα r ( t ) h ( t, t q − ) . (59)The solution of this equation is: h ( t, t q − ) ∝ h ( t q − , t q − ) = I K . (60)So the horizontal lift during [ t q − , t q ] is completely deter-mined by U q ( t, t q − ) up to an unimportant global phase: V ( t ) ∝ U q ( t, t q − ) V ( t q − ) . (61)At the end of this step, V ( t q ) ∝ g q V ( t q − ) . From Eq. (27),we could obtain the final state for a given initial state ψ ( ) ∈C : ψ ( T ) ∝ V ( T ) V † ( ) ψ ( )= p ∏ l = g p − l + V ( ) V † ( ) ψ ( )= Ω g P ( ) ψ ( )= Ω g ψ ( ) , (62)which is the encoded operation we desired. Note that the finalHamiltonian H f = ∑ s ε s Ω g P s ( ) Ω † g = H i , so our evolution iscyclic. Remark 3.
Theorem. 1 solves the problem stated in Sec. II tofind a proper path λ for given holonomy. Note that the require-ment of a fault-tolerant circuit in the implementation is crucialhere. If it is not satisfied, the horizontal lift of P ( t ) may notbe completely determined by U q at each step. The conditionthat T (G) ⊂ E local is not a very strong restriction. Indeed,it is always satisfied by stabilizer codes with d ≥ , and willgenerally be satisfied if we start with a fault-tolerant construc-tion. Also note that in principle this theorem is not restrictedto Clifford circuits that contain no π / gates. In practice, it isdifficult to build the corresponding Hamiltonians constructedin our procedure, because they are hard to represent. The fol-lowing theorem will show that, in order to make this processfault-tolerant, Clifford circuits are sufficient and preferred. B. Fault-Tolerance of the Scheme
In this section, we will discuss the fault-tolerance of thesteps to realize holonomic quantum computation as presentedabove.
Theorem 2.
Suppose we are given a fault-tolerant circuit G defined for a stabilizer code C with E ⊇ E local ⊃ T (G) . If G doesn’t contain π / , then by following the steps of the schemelisted in Sec. III A, we will get a fault-tolerant cyclic adiabaticprocess by the meaning of Def. 3.Proof. Without loss of generality, we assume an error happensat time t q (the extension of the proof to any time t is trivial).Let P ( t q ) = V ( t q ) V † ( t q ) be the projector for the code C t q .Since the circuit we follow is fault-tolerant, ∏ pl = q + g p + q − l + maps E local to a subset of E , which is a correctable error set ofour final code C (since the evolution is cyclic). It is easy tocheck that E local is a correctable error set for code C t q . Assum-ing that E t q ∈ E local is the error that happens at time t q , it canbe represented as E t q = ∑ µ c µ E t q µ where { E t q µ } is a finite setof operators that spans E local . { E t q µ } can always be chosen tosatisfy the following error-correction condition: P ( t q ) E t q † µ E t q ν P ( t q ) = d µν P ( t q ) , (63)where d µν is a diagonal matrix whose elements are either oneor zero. Those E t q µ with d µµ = have no effect on the codespace. We can always pick K ′ = n − k operators from { E t q µ } with d µµ = to form a set { E t q K ′ } . We can then constructanother set of correctable errors with linear combination ofelement in E t q K ′ : F t q α = K ′ ∑ µ = E t q µ R µα , (64)with some unitary matrix R (such a unitary matrix always ex-ists and is not unique) such that: F t q α P ( t ) F t q † α = P s α ( t ) , for all α. (65)It is easy to verify that { F t q α } still satisfies the error correctioncondition: P ( t q ) F t q † α F t q β P ( t q ) = d αβ P ( t q ) . (66)Now, E t q can be represented by: E t q = ∑ β c ′ β F t q β . Aslong as we can correct each F t q β , we can correct E t q . Sowe consider these errors individually. If no error happens,according to the Theorem. 1, the horizontal lift of P ( t ) is V ( t ) ∝ U ( t, ) V ( ) , and the final state is ψ ( T ) ∝ Ω g ψ ( ) , (67)for ψ ( ) ∈ C . The state after an error F t q β occurs is ψ ( t q ) ∝ F t q β V ( t q ) V † ( ) ψ ( ) . (68)According to Eq. (65), F t q β V ( t q ) represents an orthonormalframe of P s β ( t q ) .Next, we prove for P s β ( t ) , t > t q , that the adiabatic con-dition Eq. (50) is satisfied by this scheme. We need consideronly a single time segment [ t q , t q + ] . Again, we just treat thecase of g q + = R xm gate; the arguments for the other gates areexactly the same, since the generators of these gates are allPauli operators. For any α ≠ β , P s α ( t ) ∂H ( t ) ∂t P s β ( t ) =− i π f q + ( t ) ⋅ U X m q + ( t, t q )( P s α ( t q ) X m H ( t q ) P s β ( t q )− P s α ( t q ) H ( t q ) X m P s β ( t q )) U X m † q + ( t, t q ) , (69)where P s r ( t ) = U X m q + ( t, t q ) P s r ( t q ) U X m † q + ( t, t q ) , (70)for r = α or β . Define the index set I = { , , . . . n − k } , andtwo sets A = { j ∈ I ∣[ S j ( t q ) , g q + ] ≠ } and B = I / A . Wehave P s r ( t q ) = n − k ∏ j = I + s r j S j ( t q ) = ∏ j ∈ A I + s r j S j ( t q ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ P A s r ( t q ) ⋅ ∏ j ′ ∈ B I + s r j ′ S j ′ ( t q ) ·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ P B s r ( t q ) . (71) G is composed of gates from U , whose elements are in thenormalizer of G n , so at any stage q , S j ( t q ) ∈ G n . So we have P s α ( t q ) X m H ( t q ) P s β ( t q )= ε s β ( t q ) X m ∏ j ∈ A I − s α j S j ( t q ) P B s α ( t q ) P B s β ( t q ) P A s β ( t q ) . (72)Similarly, we have P s α ( t q ) H ( t q ) X m P s β ( t q )= ε s α ( t q ) P A s α ( t q ) P B s α ( t q ) P B s β ( t q ) ∏ j ∈ A I − s β j S j ( t q ) X m . (73) If s α j ≠ s β j for any j ∈ B , then Eq. (72) and Eq. (73) wouldbe zero, and the adiabatic condition is automatically satisfied.For those s α such that s α j = s β j for all j ∈ B , we have P s α ( t q ) X m H ( t q ) P s β ( t q )= ε s β ( t q ) X m ∏ j ∈ A I − s α j S j ( t q ) I + s β j S j ( t q ) P B s β ( t q ) , (74)and P s α ( t q ) H ( t q ) X m P s β ( t q )= ε s α ( t q ) P B s α ( t q ) ∏ j ∈ A I + s α j S j ( t q ) I − s β j S j ( t q ) X m . (75)The above two expressions are nonzero only if s β j = − s α j for all j ∈ A . Therefore, there is only one β such that P s α ˙ H ( t ) P s β ≠ and hence needs further calculation. For thatspecific s β , we have a simple relation: X m P s α ( t q ) X m = P s β ( t q ) . (76)We obtain ∥ P s α ( t ) ˙ H ( t ) P s β ( t )∥ = π f q + ( t )∥ ε s β ( t q ) X m P s β ( t q ) − ε s α ( t q ) P s α ( t q ) X m ∥ = π f q + ( t ) K ⋅ ∣ ε s α ( t q ) − ε s β ( t q )∣ . (77)So the LHS of Eq. (50) reduces to π ˙ f q + ( t ) ∣ ε s α ( t q ) − ε s β ( t q )∣ . (78)If we are in Step 3, since ∣ A ∣ is odd, we have ∣ ε s α ( t q ) − ε s β ( t q )∣ = ∣ − ∑ j ∈ A s α j ∣ ≥ , (79)and if we are in Step 4, because of our operation to break thedegeneracy by setting C b = . , we have ∣ ε s α ( t q ) − ε s β ( t q )∣ = ∣ − ∑ j ∈ A j ≠ b s α j − s α b ∣ ≥ . (80)If π ˙ f q + ( t ) ≪ is satisfied, which is always possible,then P s β ( t ) satisfies the adiabatic condition for time segment [ t q , t q + ] . The same argument can be applied to the rest of thetime segments to show that the adiabatic condition can alwaysbe satisfied by choosing appropriate functions f ( t ) .0Now, we can use the evolution equation Eq. (27): ψ β ( T ) = V s β ( T ) V † ( t q ) F t q † β F t q β V ( t q ) V † ( ) ψ ( )= V s β ( T ) V † ( t q ) V ( t q )·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ I K V † ( t q ) F t q † β F t q β × V ( t q ) V † ( t q ) V ( t q )·„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„‚„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„¶ I K V † ( ) ψ ( )= V s β ( T ) V † ( t q ) d ββ V ( t q ) V † ( t q ) V ( t q ) V † ( ) ψ ( )= V s β ( T ) V † ( ) ψ ( ) , . (81)where the third equality follows from Eq. (66). Here, V s β ( t ) is defined as V s β ( t ) = U ( t, t q ) F t q β V ( t q ) h ( t, t q ) for t > t q , (82)as the horizontal lift of P s β ( t ) given initial condition V s β ( t q ) = F t q β V ( t q ) .Again, we just focus on one time segment t ∈ [ t q , t q + ] andthe single gate R xm , since the rest are just the same: ˜ Q R xm ( t, t q ) = P s β ( t q ) Q R xm ( t, t q ) P s β ( t q )= − π f q ( t ) P s β ( t q ) X m P s β ( t q ) = . (83)So h ( t, t q ) = I K , and according to Lemma 1, V s β ( t ) = U ( t, t q ) F t q β V ( t q ) (84)is the only horizontal lift of P s β ( t ) for t > t q . FollowingEq. (81), we have ψ β ( T ) = U ( T, t q ) F t q β V ( t q ) V † ( ) ψ ( )= F Tβ U ( T, t q ) V ( t q ) V † ( ) ψ ( )= F Tβ U ( T, ) V ( ) V † ( ) ψ ( )= F Tβ Ω g ψ ( ) , (85)where F Tβ is defined as U ( T, t q ) F t q β U † ( T, t q ) , with U ( T, t q ) = ∏ pl = q + g p + q − l + . Since the circuit we follow isfault-tolerant, F Tβ ∈ E . Taking dynamic phases into account,if E t q occurs, the final state should be ψ ( T ) = ∑ β c ′ β exp (− i ∫ Tt ε s β ( t ) d t ) F Tβ Ω g ψ ( ) , (86)so the adiabatic process we propose here is fault-tolerant bythe meaning of Def. 3. Remark 4.
We now discuss some details of the adiabatic the-orem and its application to our scheme. The traditional ver-sion of the adiabatic theorem stated in [25] guarantees that the adiabatic approximation is satisfied with precision δ ≤ (cid:15) during the adiabatic evolution, if the condition sup t ∈[ ,T ] ∥ P s α ( t ) ∂∂t H ( t ) P s β ( t ) ∥ inf t ∈[ ,T ] K ( ε s α ( t ) − ε s β ( t )) ≤ (cid:15), for any α ≠ β, (87)is satisfied (note that K = k is the dimension of the codespace), which is equivalent to: sup q,t ∈[ ,T ] π ˙ f q ( t ) ≤ (cid:15) (88)in our case. However, it is known that this statement is neithersufficient nor necessary, and under certain conditions on theHamiltonian, we can obtain better results [26, 27]. Accordingto Ref. [27], for a Hamiltonian H ( ϑ ) ( ϑ = t / T ) that is analyticnear [ , ] in the complex plane, with the absolute value ofthe imaginary part of the nearest pole being γ , and the first N ≥ derivatives at boundaries equal to zero, i.e., H ( l ) ( ) = H ( l ) ( ) = for l ≤ N , if we set T = eγ N ξ d min , (89)where ξ = sup ϑ ∈[ , ] ∥ d H / d ϑ ∥ ∞ , (90)( ∥ ⋅ ∥ ∞ is standard operator norm, and d min is the minimumspectral gap) then the adiabatic approximation error satisfies δ ad ≤ (N + ) γ + e −N , (91)or equivalently, δ ad ≲ ( cT + ) γ + e − cT , (92)with c = γd eξ . This means that we can decrease the adiabaticerror exponentially with evolution time T , which is propor-tional to N . Applying this theorem to our piecewise adiabaticevolution, for the q th segment, we set T q = t q − t q − to be eγ N ξ q , where ξ q is ξ defined on the q th time segment, and f q ( t ) is chosen such that a) the boundary condition mentionedabove is satisfied, and b) H ( ϑ ) is analytic near [ , ] . The adi-abatic error for an encoded unitary operation composed of p gates can therefore be bounded by δ ad ≲ p ⋅ sup ≤ q ≤ p ( c q T q + ) γ + e − c q T q . (93)During the adiabatic process, the energy gap between theground space and any other eigenspace is lower-bounded by1, and this does not decrease with the size of the problem orthe number of levels of code concatenation. Again, we as-sume the qubits are independently coupled to the thermal en-vironment and the corresponding thermal errors are local andlow-weight during certain period of evolution time. Thoselow-weight thermal excitations will cause transitions from theground space to higher energy excited spaces. Their rate will1decrease exponentially with the existence of an energy gap[13], δ thermal ∼ O ( exp (− d min )) , while Eq. (89) shows that thetime needed to finish the process grows inversely as the cubeof the minimum gap. The system error can be bounded by thesum of these two errors [16]: δ S < δ thermal + δ ad . (94)So, qualitatively, we can make both the adiabatic error andthermal excitation exponentially small with efficient overheadin processing time. (However see the discussion in Sec. V forpossible limitations of this argument.)Theorem 2 builds an equivalence relation between a fault-tolerant encoded Clifford circuit and a fault-tolerant adiabaticprocess that gives the same encoded unitary operation. Formost stabilizer codes (e.g., Steane code, the surface code, orthe C4 code), encoded operations in the Clifford group canall be realized by such fault-tolerant circuits. Standard tech-niques, like magic state injection and distillation, can realizefault-tolerant encoded non-Clifford gates like the encoded π / and encoded Toffoli gates. Magic state distillation can be im-plemented by fault-tolerant encoded unitary gates from theClifford group. Thus, this holonomic scheme is universal forfault-tolerant quantum computation. IV. EXAMPLES
In this section, we apply the procedure developed above toconstruct adiabatic processes for specific codes. For pedagog-ical purposes, our first example realizes the encoded X for thesimplest 3-qubit repetition code. Our second example is the 7qubit Steane code. This example is of practical importance be-cause through concatenation of this simple code, fault-tolerantquantum computation can be achieved when the error rate islower than the threshold. A. 3-qubit repetition code
There are two stabilizer generators for this code, as shownbelow: 1 2 3 S Z Z IS I Z Z
The encoded X operator for this code is X = X X X .The encoded X gate can be performed by a circuit like X = R x R x R x R x R x R x , so the process takes 6 steps. The initialHamiltonian is H ( ) = − Z Z − Z Z . (95)For the first step of the adiabatic process, we have [ X , Z Z ] ≠ for t ∈ [ , t ] . So the Hamiltonian during that interval is H ( t ) = − cos ( f ( t ) π ) Z Z + sin ( f ( t ) π ) Y Z − Z Z , (96)with H ( t ) = Y Z − Z Z . In the second step, for t ∈ [ t , t ] ,the Hamiltonian is: H ( t ) = cos ( f ( t ) π ) Y Z + sin ( f ( t ) π ) Z Z − Z Z , (97)with H ( t ) = Z Z − Z Z . In the third step, we see that [ X , Z Z ] ≠ and [ X , Z Z ] ≠ , which implies thatthere might exist a P s that doesn’t satisfy the adiabatic con-dition. Actually, as shown on the left side of Fig. 3, we have {s =1, s =1}{s = 1, s = -1}{s = -1, s = 1}{s = -1, s = -1} {s = -1, s = -1}{s = -1, s = 1}{s = 1, s = -1}{s =1, s =1} FIG. 3: The variation of the energy diagram at the beginning of thethird and fourth step to break the degeneracy of space { s = , s =− } and { s = − , s = } . [ P s ={ , − } ( t ) + P s ={− . } ( t ) , X ] = , which means both P s ={ , − } ( t ) and P s ={− . } ( t ) will not satisfy the adiabaticcondition during the evolution if we do not break this degener-acy. Following the scheme in Sec. III A, we alter the Hamilto-nian at t to be H ( t ) = Z Z − . Z Z . The correspondingenergy diagram is shown on the right side of Fig. 3. Then wevary the Hamiltonian in the following way for t ∈ [ t , t ] : H ( t ) = cos ( f ( t ) π ) Z Z − sin ( f ( t ) π ) Z Y − . ( f ( t ) π ) Z Z + . ( f ( t ) π ) Y Z , (98)with H ( t ) = − Z Y + Y Z . We continue to break the de-generacy in the fourth step, since X again does not commutewith both P s ={ , − } ( t ) and P s ={ , − } ( t ) . So for t ∈ [ t , t ] ,the Hamiltonian is: H ( t ) = − cos ( f ( t ) π ) Z Y − sin ( f ( t ) π ) Z Z + . ( f ( t ) π ) Y Z + . ( f ( t ) π ) Z Z , (99)with H ( t ) = − Z Z + . Z Z , which can then be restored to H ( t ) = − Z Z + Z Z . The fifth and sixth steps are just likethe first and the second steps. The final Hamiltonian is H ( T = t ) = − Z Z − Z Z , which is equal to the initial Hamiltonian,and we obtain our geometric encoded X operation.2 Remark 5.
The encoded Z operator for this code is Z = Z Z Z . However, we cannot use our scheme to build theadiabatic process according to this circuit, since Z , Z and Z all commute with the initial Hamiltonian H ( ) . We cansee that for this simple code, there doesn’t exist an E that in-cludes { Z , Z , Z } , so the conditions for both Theorem 1 andTheorem 2 are not satisfied. B. The Steane code
Fault-tolerant quantum computation can be realizedthrough concatenation of Steane code. In our scheme, we onlyapply our scheme at the bottom level of concatenation . Forhigher levels, encoded unitary operations and error correctionare done in the usual way. By doing so, we keep the con-stant energy gap between the ground space and error spacesand thus maintain the ability to suppress low weight thermalerrors, and we can bound the weight of terms in the systemHamiltonian. One set of generators of the stabilizer group ofthe Steane code is listed below:1 2 3 4 5 6 7 S X I X X I X IS I X X I I X XS X I X I X I XS Z I Z Z I Z IS I Z Z I I Z ZS Z I Z I Z I Z
The circuits for the encoded Hadmard, encoded S and en-coded X and Z gates are all bit-wise transversal, and thusnaturally fault-tolerant. Note that each Hadmard can be de-composed into SR x S up to a global phase. The geometricrealizations of such gates are similar to that of the encoded X for the 3-qubit repetition code shown above. So in this sectionwe focus on the CNOT gate.
1. CNOT Gate
For the Steane code, the fault-tolerant encoded
CNOT → (control qubit encoded in block 1, target qubit encoded inblock 2) can be realized transversally between two blocksof qubits. We illustrate our scheme for one pair ofqubits from the two blocks, all the other operations arethe same. The initial Hamiltonian H ( ) can be writtenas − ∑ j = ( S j + S j ) , where S ij is the j th generator for the i th block. Each physical CNOT can be decomposed into ● = R zz SS R x S S S R x S up to a global phase.As we can see, there are nine gates in the circuit. The trans-formation of the first and last four single qubit gates has beendiscussed before. We will just show the Hamiltonian duringtime interval [ t , t ] when the two-qubit gate is performed.The Hamiltonian at time t can be shown to be: H ( t ) = − ∑ j = S j − S − S − Z X X X − Z X X X − Y Z Z Z − Y Z Z Z . (100)For an R zz gate acting on qubit 1 in both blocks, wehave [ Z Z , X X X X ] ≠ , [ Z Z , X X X X ] ≠ , [ Z Z , Y Z Z Z ] ≠ and [ Z Z , Y Z Z Z ] ≠ for t ∈ [ t , t ] , so the Hamiltonian during this interval can bechosen to be: H ( t ) =− ∑ j ≠ j ≠ S j − S − S − Z X X X − Z X X X − cos ( f ( t ) π ) X X X X + sin ( f ( t ) π ) Y X X X Z − cos ( f ( t ) π ) X X X X + sin ( f ( t ) π ) Y X X X Z − cos ( f ( t ) π ) Y Z Z Z − sin ( f ( t ) π ) Z X Z Z Z − . ( f ( t ) π ) Y Z Z Z − . ( f ( t ) π ) Z X Z Z Z , (101)with H ( t ) = − ∑ j ≠ j ≠ S j − S − S − Z X X X − Z X X X + Y X X X Z + Y X X X Z − Z X Z Z Z − Z X Z Z Z . (102)After all nine gates have been performed, the Hamiltonianwill be: H ( T ) = − ∑ j ≠ j ≠ ( S j + S j )− X X X X X − X X X X X − Z Z Z Z Z − Z Z Z Z Z . (103)After repeating this procedure on all 7 pairs of qubits, the3final Hamiltonian will be: H ( T ) = − X X X X X X X X − X X X X X X X X − X X X X X X X X − Z Z Z Z − Z Z Z Z − Z Z Z Z − X X X X − X X X X − X X X X − Z Z Z Z Z Z Z Z − Z Z Z Z Z Z Z Z − Z Z Z Z Z Z Z Z , (104)which is equal to H ( ) . Although the final Hamiltonianequals to the initial Hamiltonian, the maximum weight of itselements has doubled, which is not good for practical imple-mentation. Although recent results have shown how to mapsuch Hamiltonians to more physically reasonable two-bodyinteractions [28–30], it is still important to decrease the max-imum weight of the Hamiltonian terms.
2. Lowering the weight of the Hamiltonian
The maximum weight of the terms in the final Hamiltonianis 8, compared to the 4 for the initial Hamiltonian given byEq. (104). This problem may not exist for a fault-tolerantscheme based on the surface code [24], since during the pro-cess of code deformation, the weight of the stabilizer gener-ators is always bounded by 4. However, even for the Steanecode, we can lower the weight of the Hamiltonian terms dur-ing the process by using the following observation: H ( t ) = − ∑ j S j ( t ) = U L H ( t ) U † L = − ∑ j U L S j ( t ) U † L , (105)for some unitary operator U L which commutes with H ( t ) , butnot necessarily with the individual terms S j ( t ) . This meansthat the decomposition of the Hamiltonian is not unique, andwe can take advantage of this freedom. Here we set U L = CNOT . After the first three transversal CNOTs, the Hamil-tonian would become: H ( T ) = − X X X X X X − X X X X X X − X X X X X X − Z Z Z Z − Z Z Z Z − Z Z Z Z − X X X X − X X X X − X X X X − Z Z Z Z Z Z − Z Z Z Z Z Z − Z Z Z Z Z Z . (106) The maximum weight of any term is 6. This Hamiltonian isequal to the following form: H ( T ) = − X X X X X X − X X X X X X − X X X X X X − Z Z Z Z − Z Z Z Z − Z Z Z Z − X X X X − X X X X − X X X X − Z Z Z Z Z Z − Z Z Z Z Z Z − Z Z Z Z Z Z , (107)which again has maximum weight of 6. Note that the tran-sition between these two Hamiltonian representations can bearbitrarily fast, since they are equal. Then, if we perform theremaining four transversal CNOTs by an adiabatic process,the final Hamiltonian will return to the initial one representedby H ( T ) = − ∑ j = ( S j + S j ) . During the whole process, themaximum weight of terms in the Hamiltonian is reduced from8 to 6. The weight does not increase when the codes are con-catenated, since we just apply our scheme to the bottom levelof concatenation and do higher levels operations in their usualway. So we can realize HQC fault-tolerantly with maximumHamiltonian weight 6 while keeping the constant energy gap.For a small code like the Steane code, this may be the bestwe can do. For larger block codes, especially for topologicalcodes like the surface code, it is very likely that the weightof the Hamiltonian terms during the adiabatic process can bewell bounded. V. SUMMARY AND CONCLUSION
We have described a scheme for fault-tolerant HQC on astabilizer code, which takes advantage of a constant energygap during the process as well as of the intrinsic resilience ofHQC. We’ve shown that from a fault-tolerant circuit without π / gates, we can systematically construct a fault-tolerant adi-abatic process that implements the very same encoded unitaryoperation as the original circuit, with information encoded inthe ground state of the system Hamiltonian. As long as we canrealize holonomic versions of gates in the Clifford group, wecan implement fault-tolerant universal quantum computationby using magic state distillation.Holonomic single-qubit operation has been recently re-alized on trapped single Ca + ion system through adia-batic evolution [31]. Theoretical work on non-adiabatic non-abelian HQC has also been proposed [32], and correspondingexperiments have recently been realized in superconductingqubits [33] and NMR [34]. Applying our strategy to actualphysical systems will need certain techniques, like quantumgadgets [29, 30] or the digital quantum simulator [28], to buildthe many-body interactions. If the system Hamiltonian is builtin one of these effective ways, rather than being fundamental,it may dramatically change the local error model we have as-sumed. This effect needs further investigation.Our fault-tolerant HQC scheme differs from adiabatic gateteleportation (AGT) [35], and the scheme in Refs. [14, 15], in4the following ways: 1. Instead of focusing on single qubit uni-tary operations or two-qubit unitary operations, our schemeobtains holonomy through directly dragging the ground space(code space) of the system, whose path in the Grassmannmanifold forms a closed loop. 2. During the adiabatic pro-cedure, the energy spectrum of system basically remains thesame, and there always exists an energy gap between the codespace and the excited spaces, so that information is protectedfrom low weight thermal excitation by an energy gap.There are several advantages over other schemes here. Wecan reduce the low-weight error rate at the bottom level ofcode concatenation due to the existence of an energy gap,so that the frequency of error correction procedure can begreatly reduced at bottom physical level. The measurementof stabilizer generators and subsequent error correction canthemselves introduce more errors, and this is one of the rea-sons why thresholds for current fault-tolerant schemes are solow. Moreover, the number of physical qubits needed in afault-tolerant scheme is strongly dependent on the error rateat the physical level. Error rates substantially smaller thanthe threshold allow much smaller numbers of physical qubits.Eventually, our scheme may reduce the resource overheadneeded to do fault-tolerant quantum computation.On the other hand, our scheme seems naturally compatiblewith Hamiltonian-protected quantum memories [36], and hasthe potential to do fault-tolerant computation based on thosekind of memories, especially those with self-correcting ability[37, 38]. No dynamical method seems capable of manipulat-ing the topological degrees of freedom encoded in the groundspace of these memory in the presence of a system Hamilto- nian, as far as we know, since they would introduce terms thatdo not commute with the system Hamiltonian. However, ourmethod of locally deforming these Hamiltonians could poten-tially do quantum computation on such systems. We conjec-ture that during this kind of local deformation procedure, thesystem will keep its self-correction capability in a thermal en-vironment while quantum computation is implemented. Thisinteresting topic requires further investigation, and may opena new way of studying quantum computater architecture.We note that this method to construct fault-tolerant HQCis basically a serial procedure from gate to gate. For circuitswith large depth, we could investigate the possibility of par-allel operation. For large block codes or topological codes,such parallelization can be done, and is crucial in practice.We hope to apply our method to fault-tolerant schemes basedon large block codes and topological codes, which may havehigher thresholds than fault-tolerant schemes that concatenatesmall codes. Very likely, the maximum weight of the Hamilto-nian terms used to describe topological codes during adiabaticevolution will be small and well bounded. VI. ACKNOWLEGEMENTS
We would like to thank Daniel Lidar for discussion aboutthe adiabatic theorem. Y.-C.Z & T.A.B acknowledge supportfrom NSF Grants No. EMT-0829870 and No. TF-0830801,and from the ARO MURI Grant W911NF-11-1-0268. [1] M. A. Nielsen and I. L. Chuang,
Quantum Computationand Quantum Information (Cambridge University Press, Cam-bridge, 2000)[2] D. P. DiVincenzo and P. W. Shor, Phys. Rev. Lett. , 3260(1996)[3] A. Kitaev, Ann. of Phys. , 2 (2003)[4] D. Lidar and T. Brun, Quantum Error Correction (CambridgeUniversity Press, Cambridge, 2013)[5] P. Zanardi and M. Rasetti, Phys. Lett. A , 94 (1999)[6] F. Wilczek and A. Zee, Phys. Rev. Lett. , 2111 (1984)[7] P. Solinas, P. Zanardi, and N. Zangh`ı, Phys. Rev. A , 042316(2004)[8] I. Fuentes-Guridi, F. Girelli, and E. Livine, Phys. Rev. Lett. ,020503 (2005)[9] P. Solinas, M. Sassetti, P. Truini, and N. Zangh`ı, New. J. Phys , 093006 (2012)[10] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, “Quan-tum computation by adiabatic evolution,” (2000), eprintarXiv:quant-ph/0001106[11] E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, andD. Preda, Science , 472 (2001)[12] S. Jordan, E. Farhi, and P. Shor, Phys. Rev. A , 052322 (2006)[13] T. Albash, S. Boixo, D. A. Lidar, and P. Zanardi, New. J. Phys. , 123016 (2012)[14] O. Oreshkov, T. A. Brun, and D. A. Lidar, Phys. Rev. Lett. , 070502 (2009)[15] O. Oreshkov, T. A. Brun, and D. A. Lidar, Phys. Rev. A ,022325 (2009)[16] D. A. Lidar, Phys. Rev. Lett. , 160506 (2008)[17] M. S. Siu, Phys. Rev. A , 062314 (2005)[18] A. Mizel, D. A. Lidar, and M. Mitchell, Phys. Rev. Lett. ,070502 (2007)[19] S. Tanimura, D. Hayashi, and M. Nakahara, Phys, Lett. A ,199 (2004)[20] S. Tanimura, M. Nakahara, and D. Hayashi, J. Math. Phys. ,022101 (2005)[21] M. Nakahara, Geometry, Topology and Physics , 2nd ed. (Insti-tute of Physics Publishing, 2003)[22] D. Gottesman,
Stabilizer codes and quantum error correction ,Ph.D. thesis, California Institute of Technology (1997), eprintarXiv:quant-ph/9705052[23] E. Knill, Nature (London) , 39 (2005)[24] A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland,Phys. Rev. A , 032324 (2012)[25] A. Messiah, Quantum Mechanics, Vol. II (North-Holland Pub-lishing Co., Amsterdam, 1965)[26] G. A. Hagedorn and A. Joye, J. Math. Anal. and Appl. , 235(2002)[27] D. A. Lidar, A. T. Rezakhani, and A. Hamma, J. Math. Phys. , 102106 (2009) [28] H. Weimer, M. Muller, I. Lesanovsky, P. Zoller, and H. P. Buch-ler, Nat. Phys. , 382 (2010)[29] J. Kempe, A. Kitaev, and O. Regev, SIAM J. Comput. , 1070(2006)[30] R. Oliveira and B. M. Terhal, Quantum Info. Comput. , 900(2008)[31] K. Toyoda, K. Uchida, A. Noguchi, S. Haze, and S. Urabe,Phys. Rev. A , 052307 (2013)[32] E. Sj¨ovist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Jo-hansson, and K. Singh, New J. Phys. , 103035 (2012)[33] A. A. Abdumalikov Jr, J. Fink, Juliusson, K., M. Pechal, S. Berger, A. Wallraff, and S. Filipp, Nature (London) , 482(2013)[34] G. Feng, G. Xu, and G. Long, Phys. Rev. Lett. , 190501(2013)[35] D. Bacon and S. T. Flammia, Phys. Rev. Lett. , 120504(2009)[36] J. R. Wootton, J. Mod. Opt. , 1717 (2012)[37] S. Chesi, B. R¨othlisberger, and D. Loss, Phy. Rev. A , 022305(2010)[38] A. Hutter, J. R. Wootton, B. R¨othlisberger, and D. Loss, Phy.Rev. A86