Synthesis of Logical Clifford Operators Via Symplectic Geometry

2018 IEEE International Symposium on Information Theory (ISIT)

Synthesis of Logical Clifford Operators via Symplectic Geometry

Narayanan Rengaswamy∗, Robert Calderbank∗, Swanand Kadhe†, and Henry D. Pﬁster∗ ∗Department of Electrical and Computer Engineering, Duke University, Durham, North Carolina 27708, USA Email: {narayanan.rengaswamy, robert.calderbank, henry.pﬁster}@duke.edu †Department of Electrical Engineering and Computer Science, University of California, Berkeley, California 94720, USA Email: [email protected]

Abstract—Quantum error-correcting codes can be used to process of computation on the logical qubits, such efficient protect qubits involved in quantum computation. This requires assembly of choices for an operation could be useful since that logical operators acting on protected qubits be translated to each of them might interact differently with the current state physical operators (circuits) acting on physical quantum states. We propose a mathematical framework for synthesizing physical and control parameters of the system. This paper provides a circuits that implement logical Clifford operators for stabilizer proof of concept demonstration using the well-known [[6, 4, 2]] codes. Circuit synthesis is enabled by representing the desired QECC [9], [10], where our metric is to reduce the circuit depth × physical Clifford operator in CN N as a 2m × 2m binary sym- for each operator (see [11] for a detailed discussion). plectic matrix, where N =2m. We show that for an [[m,m − k]] stabilizer code every logical Clifford operator has 2k(k+1)/2 II.PHYSICALAND LOGICAL OPERATORS symplectic solutions, and we enumerate them efficiently using This section summarizes the mathematical framework for symplectic transvections. The desired circuits are then obtained by writing each of the solutions as a product of elementary quantum error correction introduced in [2], [4], [5] and des- symplectic matrices. For a given operator, our assembly of all cribed in detail in [9]. The quantum states of a single qubit of its physical realizations enables optimization over them with system are expressed as ψ = α 0 + β 1 C2, where respect to a suitable metric. Our method of circuit synthesis 1 0 | i | i | i ∈ 0 , and 1 , are called the computational basis can be applied to any stabilizer code, and this paper provides a 0 1 proof of concept synthesis of universal Clifford gates for the well- | i | i states, and α,β C satisfy α 2 + β 2 = 1 as per the Born known [[6, 4, 2]] code. Programs implementing our algorithms can ∈ | | | | be found at https://github.com/nrenga/symplectic-arxiv18a. rule [12, Chap. 3]. Any single qubit error can be expanded Index Terms—Heisenberg-Weyl group, symplectic geometry, in terms of flip, phase and flip-phase errors (on a state ψ ) Clifford group, stabilizer codes, logical operators, automorphisms described by the Pauli matrices | i 0 1 1 0 0 ι X , , Z , and Y , ιXZ = 1 0 0 1 ι −0 I.INTRODUCTION − respectively [6, Chap. 10], where ι , √ 1. The states of an The first quantum error-correcting code (QECC) was dis- − covered by Shor [1], and CSS codes were introduced by m-qubit system are described by (linear combinations of) Kro- Calderbank and Shor [2], and Steane [3]. The general class necker products of single-qubit states, and the corresponding of stabilizer codes was introduced by Calderbank, Rains, Shor (Pauli) errors are expressed as Kronecker products and Sloane [4], and by Gottesman [5]. A QECC protects E1 E2 Em, ι E1 E2 Em, m k logical qubits by embedding them into a physical system ± ⊗ ⊗···⊗ ± ⊗ ⊗···⊗ − comprising m physical qubits. QECCs can be used for the where Ei I2,X,Z,Y is the error on the i-th qubit and ∈ { } realization of fault-tolerant quantum computation [6]. For this I2 is the 2 2 identity matrix. These matrices form the × 2 purpose, any desired operation on the m k logical (protected) Heisenberg-Weyl group HWN of order 4N (also called the − m qubits must be implemented as a physical operation on the m Pauli group), where N = 2 . Note that the elements of HWN physical qubits, while preserving the code space. are interpreted as both m-qubit operators and errors. For stabilizer codes, physical realizations of Clifford opera- Given row vectors a,b Fm we define the m-qubit operator ∈ 2 tors on logical qubits can be represented by binary 1 1 m m 2m 2m D(a,b) , Xa Zb Xa Zb , (1) symplectic matrices, reducing the complexity ×dramatically ⊗···⊗ from 2m complex variables to 2m binary variables (see [7], [8] so that the group HW consists of operators D(a,b) and N ± and Section II). We exploit this fact to propose an algorithm ιD(a,b). Multiplication in HW satisfies ± N that efficiently assembles all symplectic matrices representing ′ T ′ T D(a,b)D(a′,b′)=( 1)a b +b a D(a′,b′)D(a,b). (2) physical transformations (circuits) that realize a given logical − operator on the protected qubits. This makes it possible to F2m The standard symplectic inner product in 2 is defined as optimize the choice of circuit with respect to a suitable metric, ′ ′ ′ T ′ T ′ ′ T [a,b], [a ,b ] s , a b + b a =[a,b]Ω[a ,b ] , (3) that might be a function of the quantum hardware. During the h i

F2m 0 Im TABLE I where the symplectic form in 2 is Ω = Im 0 A UNIVERSALSETOFLOGICALOPERATORSANDCORRESPONDING (see [4], [11]). Therefore, two operators D(a,b) and D(a′,b′) PHYSICAL OPERATORS. The number of 1s in Q and R directly relates to Fm ′ ′ number of gates. The N coordinates are indexed by binary vectors v ∈ 2 , commute if and only if [a,b], [a ,b ] s = 0. The isomorphism N and ev denotes the standard basis vector in C with an entry 1 in position γ : HW / ιI hF2m allowsi us to represent (up to N N 2 v and all other entries 0. Here H2k denotes the Walsh-Hadamard matrix of h± i → k multiplication by scalars) elements of HWN as binary vectors size 2 , Uk = diag(Ik,Om−k) and Lm−k = diag(Ok,Im−k). (i.e., γ(D(a,b)) , [a,b]). Logical Operator Fg Physical Operator g¯ The Clifford group CliffN consists of all unitary matrices N×N † † g C for which gD(a,b)g HW , where g is the 0 Im ⊗m N Ω= HN = H ∈ ∈ Im 0 2 Hermitian transpose of g [8]. It is the normalizer of HWN in the unitary group. We regard operators in CliffN as physical Q 0 A = a : ev 7→ e operators acting on quantum states in CN , to be implemented Q 0 Q−T Q vQ by quantum circuits. By definition, an operator g CliffN ∈ Im R induces an automorphism of HW by conjugation. Note TR= vRvT N " 0 Im# tR = diag ι that the inner automorphisms induced by matrices in HWN (R symmetric) preserve every conjugacy class D(a,b), because (2) implies ± Lm−k Uk that elements in HW either commute or anti-commute. For Gk = gk = H2k ⊗ I2m−k T N Uk Lm−k E(a,b) , ιab D(a,b), automorphism induced by g satisfies

† Ag Bg gE(a,b)g = E ([a,b]Fg) , where Fg = (4) ± Cg Dg Unitary operators gL U , where M = 2m−k, acting on ∈ M is a symplectic matrix. So it preserves symplectic inner the logical qubits are called logical operators. QECCs encode ′ ′ ′ ′ a logical state in CM into a physical state in CN . The process products, i.e., [a,b]Fg, [a ,b ]Fg s = [a,b], [a ,b ] s (see [8], h T i h i of synthesizing a logical operator gL for a QECC refers to [13]). This means that FgΩFg =Ω or equivalently finding a physical operator g¯ UN that preserves the code T T T T T T ∈ L AgBg = BgAg , CgDg = DgCg ,AgDg + BgCg = Im. space (i.e., normalizes S) and realizes the action of g on (5) the protected qubits. Two well-known methods to synthesize logical Pauli operators were described in [5] and [14]. For The fact that Fg is symplectic expresses the property that the L stabilizer codes, these imply that for all h HWM the automorphism induced by g must respect commutativity in associated physical operator h¯ HW as well.∈ Hence for all F N HWN . Let Sp(2m, 2) denote the group of symplectic 2m gL Cliff we also have g¯ ∈Cliff . The physical operators F F× M N 2m matrices over 2. Then the map φ: CliffN Sp(2m, 2) ¯ ∈ ∈F2m → h have a representation in 2 via the map γ. Using the defined by φ(g) , Fg is a homomorphism with kernel HWN , L map φ, we regard a logical Clifford operator g CliffM and every Clifford operator maps down to a matrix Fg. Hence as a symplectic matrix F Sp(2(m k), F ). For∈ stabilizer g ∈ − 2 HWN is a normal subgroup of CliffN and CliffN /HWN ∼= codes, in order to translate gL into a physical operator g¯, there Sp(2m, F2). are multiple ways to embed Fg into Fg¯ Sp(2m, F2) such A stabilizer is a subgroup S of HWN generated by com- that the corresponding g¯ operates on states∈ in CN and acts m muting Hermitian matrices [5], [6]. For a,b F , note L T 2 as desired on the states of the QECC. For each g CliffM ab ∈ 2 that E(a,b) = ι D(a,b) is Hermitian, E(a,b) = IN , our algorithm allows one to identify all such embeddings.∈ The ± ±IN ±E(a,b) and the operators 2 project onto the 1 eigenspaces idea is as follows. of E(a,b), respectively. A stabilizer S has± the additional We observe that the logical Clifford operators gL Cliff property that if it contains an operator g, then it does not ∈ M contain g. Consider a stabilizer S generated by Hermitian are uniquely defined by their conjugation relations with the − logical Paulis hL (also see [6], [8], [9]). Therefore these matrices E(ai,bi), where [ai,bi],i = 1, 2,...,k are linearly independent vectors in F2m. The stabilizer code corresponding relations can be directly translated to their physical equivalents 2 g¯ and h¯, i.e., gLhL(gL)† = (h′)L HW g¯h¯g¯† = h¯′ to S is the subspace V (S) fixed pointwise by S, i.e., ∈ M ⇒ ∈ HWN as well. Using the relation in (4), these conditions are N V (S)= ψ C : g ψ = ψ g S . (6) translated into linear constraints on Fg¯. Then, linear constraints {| i∈ | i | i ∀ ∈ } that require Fg¯ to normalize S are added. The set of all IN +E(a1,b1) IN +E(ak,bk) F Observe that the operator 2 2 Fg¯ Sp(2m, 2) that satisfy these constraints identify all ×···× m−k ∈ projects onto V (S), and that dim V (S) = 2 , M. embeddings of Fg into Sp(2m, F2). After we obtain Fg¯, we Such a code encodes m k logical qubits into m physical synthesize a corresponding physical operator g¯ by factoring − qubits. Hence an [[m, m k]] QECC is an embedding of Fg¯ into elementary symplectic matrices from Table I. Note m−k − m a 2 -dimensional Hilbert space into a 2 -dimensional that there are multiple circuits g¯ for a given Fg¯. In the next Hilbert space. Note that all quantum codes are not necessa- section, we carry out the process of finding universal Clifford rily stabilizer codes (see [4]). Logical qubits are commonly gates for the well-known [[6, 4, 2]] CSS code [9], [10], and then referred to as protected qubits or encoded qubits. discuss the general case.

792 2018 IEEE International Symposium on Information Theory (ISIT)

III.LOGICAL OPERATOR SYNTHESIS:THE [[6, 4, 2]] CODE In general, to define valid logical Pauli operators, it can X Z The [6, 5, 2] single-parity check code is generated by be observed that the matrices GC/C⊥ ,GC/C⊥ must satisfy C T X Z Z HC 1 1 0 0 0 0 GC/C⊥ GC/C⊥ = Im−k and GC = Z must form GC/C⊥ HC , 1 0 1 0 0 0 GC = ; GC/C⊥   , (7) another generator matrix for the (classical) code . It can be G ⊥ 1 0 0 1 0 0 C C/C verified that the above matrices satisfy these conditions. 1 0 0 0 1 0 ¯   2) Logical Phase Gate: The phase gate g¯ = P1 on the first   where the parity-check matrix is HC = [1 1 1 1 1 1]. logical qubit (i.e., the physical implementation) is defined by The rows h , for i = 1, 2, 3, 4, of G ⊥ generate all coset i C/C Y¯ if j = 1, representatives for ⊥ in . The CSS construction [2], [3], [6] P¯ X¯ P¯† = j , P¯ Z¯ P¯† = Z¯ j = 1, 2, 3, 4. 1 j 1 ¯ 1 j 1 j provides a [[6, 4, 2]]CstabilizerC code spanned by the states (Xj if j = 1, ∀ Q 6 (14) 4 4 1 1 One can express P¯1 in terms of the physical Paulis Xt,Zt (000000) + xihi + (111111) + xihi , as follows. The condition P¯ X¯ P¯† = Y¯ implies P¯ must √2 + √2 + 1 1 1 1 1 i=1 i=1 transform X¯ = X X into Y¯ , ιX¯ Z¯ = ιX X Z Z = X X (8) 1 1 2 1 1 1 1 2 2 6 F X1(ιX2Z2)Z6 = X1Y2Z6. Similarly, the other conditions where xi 2, i = 1, 2, 3, 4. Let X t and Zt denote the X ∈ ¯ ¯ and Z operators, respectively, acting on the t-th physical qubit. imply that all other Xjs and all Zjs must remain unchanged. Then the physical operators Direct inspection of these conditions yields the circuit given below. First we find an operator which transforms X2 to Y2 gX gZ = X1X2X3X4X5X6, = Z1Z2Z3Z4Z5Z6 (9) and leaves other Paulis unchanged; this is P2, the phase gate on the second physical qubit. Then we find an operator that generate the stabilizer group S that determines . † Q transforms Y2 into Y2Z6, which is CZ26 as X2CZ26X2 = A. Logical Operators for Protected Qubits † X2Z6 and ZiCZ26Zi = Zi,i = 1, 2,..., 6. Here CZ26 is the We construct logical Clifford operators by synthesizing controlled-Z gate on physical qubits 2 and 6. But this also physical operators g¯ on the physical qubits. Since the operator transforms X6 into Z2X6 and hence the circuit CZ26P2 does X g¯ preserves , conjugation by g¯ must preserve the stabilizer not fix the stabilizer g . Hence we include P6 so that the full Q ⊥ X Z ¯ S and its normalizer S in HWN , i.e., the dual of S with circuit P6CZ26P2 fixes g , g and also realizes P1. respect to the symplectic inner product [4]. We note that g¯ 2 P need not commute with every element of the stabilizer . S x1 P ≡ | iL 1) Logical Paulis: Let x L be the logical state defined 6 P | Fi4 by x = [x1,x2,x3,x4] 2 in (8). Then the generating L L ∈ set Xi ,Zi HW24 ,i = 1, 2, 3, 4 for the logical Pauli We now describe how this same circuit can be synthesized { ∈ } A B operators are defined by the actions via symplectic geometry. Let F = be the symplectic C D x 1, if j = i ¯ XL x = x′ , where x′ = i ⊕ matrix corresponding to P1. The conditions imposed in (14) i | iL | iL j x , if j = i on logical qubit operators X¯ ,j = 1, 2, 3, 4 give, as per (4), ( j 6 j and ZL x =( 1)xi x . (10) [110000, 000000]F = [110000, 010001] (X X X Y Z ) i | iL − | iL 1 2 7→ 1 2 6 L We denote the physical operators corresponding to and (e1 + e2)A = e1 + e2, (e1 + e2)B = e2 + e6, Xi ⇒ L ¯ ¯ X , ⊥ Zi as Xi and Zi, respectively. Set GC/C⊥ GC/C and set and (e1 + ei)A = e1 + ei, (e1 + ei)B = 0, i = 3, 4, 5. ¯ 0 1 0 0 0 1 Similarly, the conditions imposed on Zj,j = 1, 2, 3, 4 give Z 0 0 1 0 0 1 G ⊥ , . (11) (ei + e6)C = 0, (ei + e6)D = ei + e6, i = 2, 3, 4, 5. C/C 0 0 0 1 0 1 ¯ 0 0 0 0 1 1 Although it is sufficient for P1 to just normalize S, we can   always require that the physical operator commute with every   We use the rows of these two matrices to define logical Pauli element of S, i.e., centralize S (see [11, Theorem 28]). operators X¯i, Z¯i,i = 1, 2, 3, 4 as follows (see [11, Section V]). (e + ... + e )A = e + ... + e =(e + ... + e )D, X¯ = X X Z¯ = Z Z 1 6 1 6 1 6 1 1 2 1 2 6 (e + ... + e )B = 0 =(e + ... + e )C. X¯ = X X Z¯ = Z Z 1 6 1 6 2 1 3 2 3 6 . (12) X¯3 = X1X4 Z¯3 = Z4Z6 We observe that one solution is F = TB (see Table I), where ¯ ¯ X4 = X1X5 Z4 = Z5Z6 0 0 0 0 0 0 These operators commute with every element of the stabilizer 0 1 0 0 0 1   S and satisfy, as required, 0 0 0 0 0 0 I6 B B , BP = F = . 0 0 0 0 0 0 ⇒ 0 I6 Z¯ X¯ if i = j,   X¯ Z¯ = j i . (13) 0 0 0 0 0 0 i j −¯ ¯   (ZjXi if i = j 0 1 0 0 0 1 6    

793 2018 IEEE International Symposium on Information Theory (ISIT)

T The resulting physical operator P¯ = diag ιvBP v satisfies A 0 1 transformation F . We identify the solution F = −T , ¯ 0 A P1 = P6CZ26P2 and hence coincides with the above circuit. where Henceforth, we refer to g¯ itself as the logical operator. 1 0 0 0 0 0 1 0 1 0 0 1 3) Logical Controlled-Z (CZ): The logical operator CZ 12 0 1 0 0 0 0 0 1 1 0 0 1 is defined by its action on the logical Paulis as 1 1 1 0 0 0 0 0 1 0 0 0 A = ,A−T = . ¯ ¯ 0 0 0 1 0 0 0 0 0 1 0 0 X1Z2 if j = 1,     ¯ † ¯ ¯ 0 0 0 0 1 0 0 0 0 0 1 0 CZ12XjCZ12 = Z1X2 if j = 2, ,      1 1 0 0 0 1 0 0 0 0 0 1 X¯j if j = 1, 2     6 The action of on logical qubits is related to the ¯ † ¯ CNOT2→1 CZ12ZjCZ12 = Zj j = 1, 2, 3, 4. (15)  ∀ action on physical qubits through the generator matrix GC/C⊥ . We first express the logical operator CZ , on the first two The map v vA fixes the code (i.e., ev = v 12 7→ C X | i 7→Z evA = vA fixes and hence its stabilizers g and g ) logical qubits, in terms of the physical Pauli operators Xt,Zt. and induces| i a linearQ transformation on the coset space / ⊥ CZ12 CZ12 C C X¯ = X X X X Z Z Z¯ = Z Z Z Z (which defines the CSS state). The action K on logical qubits 1 1 2 7−→ 1 2 3 6 1 2 6 7−→ 2 6 . CZ12 CZ12 (bits) is related to the action A on physical qubits (bits) by X¯2 = X1X3 X1X3Z2Z6 Z¯2 = Z3Z6 Z3Z6 X X K G ⊥ = G ⊥ A and we obtain 7−→ 7−→ · C/C C/C · ¯ ¯ ¯ ¯ X3, X4, Z3, Z4 are left unchanged by CZ12. As with the 1 0 0 0 phase gate, we translate these conditions into linear equations 1 1 0 0 K = involving the constituents of the corresponding symplectic 0 0 1 0 transformation F . We again obtain F = TB, where 0 0 0 1   0 0 0 0 0 0 as desired. The circuit shown on the left below implements the 0 0 1 0 0 1 operator e e on physical qubits, where e is a standard   v vA v 0 1 0 0 0 1 basis vector7→ in CN as defined in Table I. The circuit on the B , B = . CZ 0 0 0 0 0 0 right implements e e , where x F4, i.e. CNOT .   x xK 2 2→1 0 0 0 0 0 0 7→ ∈   1 0 1 1 0 0 0 x2   2 | iL  T vBCZv The physical operator CZ12 = diag ι commutes 3 ≡ x1 L with the stabilizer gZ but not with gX ; it takes X⊗6 to X⊗6. 6 | i − This is remedied through post multiplication by Z6, resulting in the circuit obtained by Chao and Reichardt [10]: We note that [15] discusses codes and operators where A is a permutation matrix corresponding to an automorphism of . 2 x Remark: To implement CNOT we can also use the identityC | 1iL 2→1 3 ≡ x 2 2 6 Z | 2iL = 1 1 H¯1 H¯1 4) Logical Controlled-NOT (CNOT): The logical operator

CNOT2→1, where logical qubit 2 controls 1, is defined by where H¯1 is the targeted Hadamard operator (synthesized ¯ ¯ below). However, this construction might require more gates. † X1X2 if j = 2, ¯ CNOT X¯ CNOT = , 5) Logical Targeted Hadamard: The Hadamard gate H1 on 2→1 j 2→1 ¯ Xj if j = 2 the first logical qubit is defined by the actions ( 6 † Z¯ Z¯ if j = 1, ¯ ¯ CNOT Z¯ CNOT = 1 2 (16) ¯ ¯ ¯ † Zj if j = 1, ¯ ¯ ¯ † Xj if j = 1, 2→1 j 2→1 ¯ H1XjH1 = , H1ZjH1 = . Zj if j = 1. X¯ if j = 1, Z¯ if j = 1, ( 6 ( j ( j 6 6 (17) We approach synthesis via symplectic geometry, and express As before, we translate these conditions into linear equations the operator CNOT in terms of the physical operators 2→1 involving the constituents of the corresponding symplectic X ,Z as shown below. t t transformation F . We identify one possible solution as ¯ 2→1 ¯ 2→1 X1 = X1X2 X1X2 Z1 = Z2Z6 Z2Z3 1 0 0 0 0 0 0 0 0 0 0 0 7−→2→1 7−→2→1 . X¯2 = X1X3 X2X3 Z¯2 = Z3Z6 Z3Z6 1 0 0 0 0 0 0 1 0 0 0 1 7−→ 7−→ 0 0 1 0 0 0 0 0 0 0 0 0 ¯ ¯ ¯ ¯ A = , B = , X3, X4, Z3, Z4 are again left unchanged by CNOT2→1. As 0 0 0 1 0 0 0 0 0 0 0 0     before, we translate these conditions into linear equations 0 0 0 0 1 0 0 0 0 0 0 0     involving the constituents of the corresponding symplectic 1 1 0 0 0 1 0 1 0 0 0 1        

794 2018 IEEE International Symposium on Information Theory (ISIT)

1 1 0 0 0 0 1 1 0 0 0 1 6) Express the operator g¯ as a sequence of physical Clifford 1 1 0 0 0 0 0 0 0 0 0 1 gates to obtain the desired circuit for g¯. 0 0 0 0 0 0 0 0 1 0 0 0 C = , D = . In step 3 one can obtain all valid solutions F as follows: 0 0 0 0 0 0 0 0 0 1 0 0     Combine all linear conditions on F obtained in step 2 to 0 0 0 0 0 0 0 0 0 0 1 0     obtain a system of equations UF = V . Then vectorize both 0 0 0 0 0 0 0 0 0 0 0 1 sides to get (I U) vec(F ) = vec(V ). Perform Gaussian    (18) 2m     elimination on the⊗ augmented matrix [(I U) , vec(V )]. 2m ⊗ The unitary operation corresponding to this solution commutes If ℓ is the number of non-pivot variables in the row-reduced ℓ with each stabilizer element. Another solution for H¯1 which echelon form, then there are 2 solutions to the linear system. fixes Z⊗6 but takes X⊗6 (111111, 000000) to Y ⊗6 All such solutions that satisfy F ΩF T = Ω are feasible (111111, 111111) is given by↔ just changing B above to ↔ symplectic solutions for g¯. In [11] we give a more elegant and efficient algorithm for this task using symplectic transvections. 0 0 0 0 0 1 We explicitly show that for an [[m, m k]] stabilizer code every 0 1 0 0 0 0 logical Clifford operator has 2k(k+1)−/2 symplectic solutions. 0 0 0 0 0 1 B = . (19) 0 0 0 0 0 1 IV. CONCLUSION   0 0 0 0 0 1 In this work we have used symplectic geometry to pro-   1 0 1 1 1 1 pose a systematic framework for synthesizing logical Clifford     operators for any stabilizer code. Our algorithm provides as a However, for both these solutions the resulting symplectic solution all feasible symplectic matrices, which are then trans- transformation does not correspond to any of the elementary formed into circuits by decomposing them into elementary forms in Table I. Hence the unitary needs to be determined forms. This decomposition is not unique, and in future work by expressing F as a sequence of elementary transformations we will optimize for circuit complexity and fault-tolerance. and then multiplying the corresponding unitaries. An algorithm for this is given in [13] (see [11, Theorem 23]). For the ACKNOWLEDGMENT solution (18), we verified our matrix with the circuit in [10]: The authors would like to thank Jungsang Kim and Jianfeng Lu for helpful discussions. S. Kadhe would like to thank Alex 1 H H X Sprintson for his continued support. 2 H REFERENCES 6 Z [1] P. W. Shor, “Scheme for reducing decoherence in quantum computer memory,” Phys. Rev. A, vol. 52, no. 4, pp. R2493–R2496, 1995. [2] A. R. Calderbank and P. W. Shor, “Good quantum error-correcting codes B. Proposed Algorithm and Discussion exist,” Phys. Rev. A, vol. 54, pp. 1098–1105, Aug 1996. [3] A. M. Steane, “Simple quantum error-correcting codes,” Phys. Rev. A, The synthesis of logical Paulis by Gottesman [5] and by vol. 54, no. 6, pp. 4741–4751, 1996. Wilde [14] exploits symplectic geometry over the binary [4] A. Calderbank, E. Rains, P. Shor, and N. Sloane, “Quantum error field. Building on their work we have demonstrated, using correction via codes over GF(4),” IEEE Trans. Inform. Theory, vol. 44, pp. 1369–1387, Jul 1998. the [[6, 4, 2]] code as an example, that symplectic geometry [5] D. Gottesman, Stabilizer codes and quantum error correction. PhD provides a systematic framework for synthesizing physical thesis, California Institute of Technology, 1997. implementations of any logical operator in the logical Clifford [6] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information. Cambridge university press, 2010. group CliffM . The algorithm comprises the following steps: [7] A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, 1) Determine the target logical operator g¯ by specifying its “Quantum Error Correction and Orthogonal Geometry,” Phys. Rev. Lett., vol. 78, no. 3, pp. 405–408, 1997. conjugation relations with the logical Pauli operators [8]. [8] D. Gottesman, “An Introduction to Quantum Error Correction and Fault- 2) Transform the above relations into linear equations on Tolerant Quantum Computation,” arXiv preprint arXiv:0904.2557, 2009. the target symplectic transformation F . Add conditions [9] D. Gottesman, “A Theory of Fault-Tolerant Quantum Computation,” arXiv preprint arXiv:quant-ph/9702029, 1997. for normalizing the stabilizer S. [10] R. Chao and B. W. Reichardt, “Fault-tolerant quantum computation with 3) Derive a feasible solution for F (satisfies F ΩF T =Ω). few qubits,” arXiv preprint arXiv:1705.05365, 2017. 4) Factor F into a product of elementary symplectic trans- [11] N. Rengaswamy, R. Calderbank, S. Kadhe, and H. D. Pfister, “Synthesis of Logical Clifford Operators via Symplectic Geometry,” arXiv preprint formations listed in Table I, possibly using the algorithm arXiv:1803.06987, 2018. given in [13] (see [11, Algorithm 3]), and compute g¯. [12] M. M. Wilde, Quantum Information Theory. Cambridge University 5) Check for conjugation of g¯ with the stabilizer genera- Press, 2013. [13] T. Can, “An algorithm to generate a unitary transformation from loga- tors and for the conditions derived in step 1. If some rithmically many random bits.” Research Independent Study, Preprint, signs are incorrect, post-multiply by an element from 2017. HW as necessary to satisfy these conditions (use [6, [14] M. M. Wilde, “Logical operators of quantum codes,” Phys. Rev. A, N vol. 79, no. 6, p. 062322, 2009. Proposition 10.4]). Note that every Pauli operator in [15] M. Grassl and M. Roetteler, “Leveraging automorphisms of quantum HWN induces the symplectic transformation I2m, so codes for fault-tolerant quantum computation,” in Proc. IEEE Int. Symp. post-multiplication does not change the target matrix. Inform. Theory, pp. 534–538, Jul 2013.

795