Synthesis of Logical Operators for Quantum Computers using Stabilizer Codes

Transcription

1 Synthesis of Logical Operators for Quantum Computers using Stabilizer Codes Narayanan Rengaswamy Ph.D. Student, Information Initiative at Duke (iid) Duke University, USA Seminar, Department of Electrical Engineering Indian Institute of Technology, Madras, India arxiv: April 26, 2018 Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

2 Joint Work Robert Calderbank Trung Can Narayanan Rengaswamy Henry Pfister Swanand Kadhe Logical Clifford Operators in Q. Computing Jianfeng Lu Jungsang Kim arxiv: / 47

3 Problem: Operations on Encoded Qubits unreliable circuit initial logical state x L arbitrary logical operation x L desired final state QECC encode QECC decode relevant physical operation ψ x ψ x QECC: Quantum Error-Correcting Codes Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

4 Problem: Operations on Encoded Qubits unreliable circuit initial logical state QECC encode x L arbitrary logical operation Need to translate for given QECC x L QECC decode desired final state relevant physical operation ψ x ψ x We do this for logical Clifford operations on stabilizer QECCs QECC: Quantum Error-Correcting Codes Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

5 In this talk... Synthesis of logical Pauli operators for CSS codes Two popular algorithms in the literature: [Got97b; Wil09]. We provide a closely-related but classical coding-theoretic perspective. Synthesis of logical Clifford operators for stabilizer codes Methods seem to exist only for particular QECCs and operations, e.g., [Got97a; Fow+12; GR13; CR17]. We propose a systematic framework using symplectic geometry. We efficiently enumerate all symplectic matrices representing a given operator. We highlight some technical results from our paper. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

6 Overview 1 Mathematical Setup for Quantum Computing 2 Logical Pauli Operators for CSS Codes 3 Logical Clifford Operators for Stabilizer Codes 4 General Algorithms and Results Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

7 Kronecker Products of Matrices Given a p q matrix A = [a ij ] and a r s matrix B = [b kl ], the Kronecker product A B is defined by a 11 B a 1q B A B... a p1 B a pq B pr qs Lemma Given matrices A, B, A, B, we have (A B)(A B ) = (AA ) (BB ), and in general (A 1 A m )(B 1 B m ) = (A 1 B 1 ) (A m B m ). Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

8 Pure States Qubit: Mathematically, it is a 2-dimensional Hilbert space over C. Pure state: ψ = α 0 + β 1, with α, β C and α 2 + β 2 = 1. Example (m = 2 qubits) : 0 1 = 1 0 = [ ] 1 0 [ ] 0 1 [ ] 0 = 1 [ ] 1 = 0 Note that C N = C 2m = C 2 C 2 (m times). N = 2 m = 01, = Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

9 Heisenberg-Weyl Group HW N The Heisenberg-Weyl (or Pauli) group for a single qubit: HW 2 ι κ {I 2, X, Z, Y }, ι 1, κ {0, 1, 2, 3}. Bit-Flip: Phase-Flip: Bit-Phase Flip: [ ] 0 1 X X v = v 1, v {0, 1}. 1 0 [ ] 1 0 Z Z v = ( 1) v v. 0 1 [ ] 0 ι Y = ιxz. XZ = ZX. ι 0 {I 2, X, Z, Y } forms an orthonormal basis for the real vector space of Hermitian operators on C 2, under the trace inner product. For m Qubits: HW N Kronecker products of m HW 2 matrices (N = 2 m ). Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

10 Binary Representation of HW N (N = 2 m ) E.g.: (XZ X Z XZ I 2 ) = XZ 1 X 0 Z 1 XZ 0 I 2 1 = Definition Given binary m-tuples a = (a 1,..., a m ), b = (b 1,..., b m ) define the matrix D(a, b) X a 1 Z b 1 X am Z bm C N N. E.g.: D(a, b) = D(11010, 10110) = XZ X Z XZ I 2 Y 1 X 2 Z 3 Y 4. D(a, b) v = ( 1) vbt v + a D(11010, 10110) = HW N Group: All matrices of the form ι κ D(a, b), κ {0, 1, 2, 3}. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

11 Isomorphism: γ(d(a, b)) [a, b] Property: D(a, b)d(a, b ) = ( 1) a b T D(a + a, b + b ). Multiplication of HW N elements addition of binary vectors. Property: D(a, b)d(a, b ) = ( 1) a b T +b a T D(a, b )D(a, b). Symplectic Inner Product [Cal+98]: For vectors [a, b], [a, b ] F 2m 2, define where Ω = [a, b], [a, b ] s a b T + b a T = [a, b] Ω [a, b ] T, [ ] 0 Im is the symplectic form in F I m 0 2m 2. D(a, b), D(a, b ) HW N commute iff [a, b], [a, b ] s = 0. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

12 Stabilizer Groups (N = 2 m ) k-dimensional stabilizer: commutative subgroup S HW N generated by linearly independent Hermitian operators E(a j, b j ) ι abt D(a j, b j ), j = 1,..., k. Example: 2-dimensional subgroup of HW 2 6(m = 6) generated by g X = E(111111, ) = X 6 g Z = E(000000, ) = Z 6. and Generator Matrix (using the isomorphism γ): G = [a j, b j ] j=1,...,k. Example: G = [ ]. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

13 Stabilizer Codes (N = 2 m ) k-dimensional stabilizer: commutative subgroup S HW N generated by linearly independent Hermitian operators E(a j, b j ) ι abt D(a j, b j ), j = 1,..., k. [[m, m k, d]] Stabilizer Code: The 2 m k dimensional subspace V (S) jointly fixed by all elements of the stabilizer S. { } V (S) ψ C N g ψ = ψ g S. The [[6, 4, 2]] Code: S g X = X 6 = E(r, 0), g Z = Z 6 = E(0, r), r = [111111]. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

14 Example: The [[6, 4, 2]] CSS Code Instance of Calderbank-Shor-Steane (CSS) construction [CS96; Ste96]. Built from the classical [6, 5, 2] single-parity-check code C. Generator matrix: G C = [ HC G C/C r ] h 1 = h 2 h 3 = h V (S): stabilizer code spanned by the (basis) states (x = (x 1, x 2, x 3, x 4 ) F 4 2) x L ψ x (000000) + x j h j (111111) + x j h j. Note: V (S) is preserved by S = X 6 = E(r, 0), Z 6 = E(0, r). j=1 j=1 Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

15 Overview 1 Mathematical Setup for Quantum Computing 2 Logical Pauli Operators for CSS Codes 3 Logical Clifford Operators for Stabilizer Codes 4 General Algorithms and Results Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

16 Recollect: Operations on Encoded Qubits [[6, 4, 2]] QECC encode x L logical HW operation x L relevant physical operation ψ x ψ x Notation: g from Heisenberg-Weyl Need to translate for the [[6, 4, 2]] QECC [[6, 4, 2]] QECC decode g L is the logical operation and ḡ is the corresponding physical operation. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

17 Logical Pauli Operators X L j, Z L j HW 2 4 Defining Properties: X L j x L = x L, where x i = and Z L j x L = ( 1) x j x L. These operators also satisfy X L i Z L j = x 1 x 2 x 3 x 4 L ψ x 1 2 Observe: (000000) + { x j 1 x i { Z L j X L i if i = j, Z L j X L i if i j. 4 x j h j j= (111111) +, if i = j, if i j 4 x j h j. Applying Xj L to x L Adding/removing h j in each term of ψ x. Applying Zj L to x L Multiplying ψ x by ( 1) iff h j present in it. j=1 Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

18 Synthesizing X L j and Z L j Synthesis refers to finding physical operators X j and Z j such that: X j, Z j U N act on ψ x and realize action of Xj L, Z j L (resp.) on x L. { X j, Z j satisfy the commutation relations: Xi Zj = Z j Xi if i = j,. Z j Xi if i j X j, Z j normalize the stabilizer S so that, for g S g S s.t. X j ψ x = X j g ψ x = ( X j g X j ) X j ψ x = g X j ψ x (similarly for Z j ). In other words, X j, Z j preserve the code subspace. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

19 Synthesizing X L j x 1 x 2 x 3 x 4 L ψ x 1 2 and Z L j (000000) + 4 x j h j j= (111111) + 4 x j h j. j=1 Generator matrices G X C/C and G Z C/C satisfying G X C/C (G Z C/C ) T = Im k : G X C/C = and G Z C/C = X j and Z j specified resp. by the j th row of G X C/C (h j ) and G Z C/C (h j ). X j D(h j, 0), Zj D(0, h j), X i Zj = { Z j X i if i = j,, Z X j S X j = S, Z j S Z j = S. j Xi if i j Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

20 Overview 1 Mathematical Setup for Quantum Computing 2 Logical Pauli Operators for CSS Codes 3 Logical Clifford Operators for Stabilizer Codes 4 General Algorithms and Results Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

21 The Clifford Group Cliff N U N Cliff N N UN (HW N ): all g U N for which ghw N g = HW N. Cliff N = HW N, H, P, CNOT or CZ Cliff N HW N S I N Gate Unitary Matrix Action on Paulis [ ] Hadamard H HX = ZH HZ = XH Phase Controlled-NOT Controlled-Z P C 1 2 CZ 12 [ ] ι [ ] I2 0 0 X [ ] I2 0 0 Z PX = YP PZ = ZP C 1 2 (X I 2 ) = (X X )C 1 2 CZ 12 (X I 2 ) = (X Z)CZ 12 Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

22 Cliff N and Symplectic Geometry U N CliffN = {g U N ghw N g = HW N } Aut(HW N ). Cliff N HW N S I N Symplectic Representation Define E(a, b) ι abt D(a, b). If g Cliff N then [ ] ge(a, b)g Ag B = ±E ([a, b]f g ), where F g = g is symplectic, i.e., satisfies F g ΩF T g = Ω = C g [ ] 0 Im. I m 0 F g ΩF T g = Ω: [a, b], [a, b ] s = [a, b]f g, [a, b ]F g s. D g Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

23 Example: The Controlled-Z Gate [ ] I2 B g = CZ 12, F g = g = I 2 1 0, B g = 0 1 [ ] Symplectic Representation: ge(a, b)g = ±E ([a, b]f g ). g(x I 2 )g = ge(10, 00)g = E ([10, 00]F g ) = E(10, 01) = X Z (or) CZ 12 (X I 2 ) = (X Z)CZ 12. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

24 The Symplectic Group Sp(2m, F 2 ) [Cal+98] Recollect that D(a, b)d(a, b ) = ( 1) [a,b],[a,b ] s D(a, b )D(a, b). Hence, if g is a D(a, b) then F g = I 2m, since ge(a, b)g = ±E(a, b). Sp(2m, F 2 ) {F F 2m 2m 2 F ΩF T = Ω}. Map φ: Cliff N Sp(2m, F 2 ), φ(g) F g, is a group homomorphism with kernel HW N, i.e., φ(g) = I 2m for g HW N. Map γ : HW N F 2m 2, γ(d(a, b)) [a, b], is also a group homomorphism with kernel ι κ I N, i.e., γ(h) = [0, 0] for h ι κ I N. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

25 Elementary Symplectic Matrices Symplectic Matrix F g Physical Operator g Clifford Element Ω = A Q = [ 0 ] Im I m 0 [ Q 0 ] 0 Q T [ ] Im R T R = 0 I m with R symmetric [ ] Lm k U G k = k U k L m k U k = diag (I k, O m k ) L m k = diag (O k, I m k ) H N = H m 2 Full Hadamard a Q : v vq ( ) t R = diag ι vrv T g k = H 2 k I 2 m k CNOTs, Permutations Phase (P), Controlled-Z (CZ) Partial Hadamards Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

26 Recollect: Operations on Encoded Qubits [[6, 4, 2]] QECC encode x L Notation: g from Clifford logical Clifford operation Need to translate for the [[6, 4, 2]] QECC x L relevant physical operation ψ x ψ x [[6, 4, 2]] QECC decode g L is the logical operation and ḡ is the corresponding physical operation. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

27 Synthesis Problem Find ḡ U N for each g L {P1 L, CZL 12, CNOT L 2 1, H1 L } such that: ḡ realizes the action of g L on the encoded qubits. ḡ acts on HW 2 6 the same way g L acts on HW 2 4 (under conjugation): g L h L = (h ) L g L ḡ h = h ḡ. ḡ centralizes S = X 6, Z 6 (commutes with every element of S). stronger condition than normalize, but is always possible (see paper). Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

28 Synthesizing g L = CNOT L 2 1: flip x 1 if x 2 = 1 x 1 x 2 x 3 x 4 L ψ x 1 C c C c + x G X C/C = 1 2 c + c C Implementing CNOT L on x L x x K, K = c C 4 x j h j. Then we find that K G X = G X Q, where Q fixes the code C, i.e., C/C C/C c + xkg X C/C = c + xg X C/C Q = ) (c + xg X C/C Q. c C c C j=1 ḡ = x 1 L x 2 L = g L Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

29 Synthesizing g L = CNOT L 2 1: flip x 1 if x 2 = 1 x 1 x 2 x 3 x 4 L ψ x 1 C c C c + x G X C/C = 1 2 c + c C Implementing CNOT L on x L x x K, K = c C 4 x j h j. Then we find that K G X = G X Q, where Q fixes the code C, i.e., C/C C/C c + xkg X C/C = c + xg X C/C Q = ) (c + xg X C/C Q. c C c C j=1 ḡ = x 1 L x 2 L = g L Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

30 Recollect Some Identities... PZ = ZP PX = YP Z P = P Z X P = P Y CZ 12 (X I 2 ) = (X Z)CZ 12 X = X Z CZ 12 (Z I 2 ) = (Z I 2 )CZ 12 Z = Z CZ 12 (Y I 2 ) = (Y Z)CZ 12 Y = Y Z Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

31 Synthesizing g L = P L 1 Phase gate on the 1st logical qubit: (translate P L 1 P 1 ) Definition : P 1 X j P 1 {Ȳj if j = 1, X j if j 1,, P 1 Z j P 1 Z j j = 1, 2, 3, 4. P1 X 1 = X 1 X 2 X 1 = X 1 Y 2 Z 6 P1 Z1 = Z 2 Z 6 Z 1 = Z 2 Z 6 P X 2 = X 1 X 1 3 X 2 = X 1X 3 P Z 2 = Z 3 Z 1 6 Z 2 = Z 3Z 6 P1 X 3 = X 1 X 4 X 3 = X 1 X 4 P1 Z3 = Z 4 Z 6 Z 3 = Z 4 Z 6 P X 4 = X 1 X 1 5 X 4 = X 1X 5 P Z 4 = Z 5 Z 1 6 Z 4 = Z 5Z 6. P 1 = 6 2 x 1 L P = P1 L Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

32 Synthesizing g L = P L 1 Phase gate on the 1st logical qubit: (translate P L 1 P 1 ) Definition : P 1 X j P 1 {Ȳj if j = 1, X j if j 1,, P 1 Z j P 1 Z j j = 1, 2, 3, 4. P X 1 = X 1 X 1 2 X 1 = X 1Y 2 Z 6 P Z 1 = Z 2 Z 1 6 Z 1 = Z 2Z 6 P1 X 2 = X 1 X 3 X 2 = X 1 X 3 P1 Z2 = Z 3 Z 6 Z 2 = Z 3 Z 6 P X 3 = X 1 X 1 4 X 3 = X 1X 4 P Z 3 = Z 4 Z 1 6 Z 3 = Z 4Z 6 P1 X 4 = X 1 X 5 X 4 = X 1X 5 P1 Z 4 = Z 5 Z 6 Z 4 = Z 5Z 6. P 1 = 6 2 P x 1 L P = P1 L Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

33 Synthesizing g L = P L 1 Phase gate on the 1st logical qubit: (translate P L 1 P 1 ) Definition : P 1 X j P 1 {Ȳj if j = 1, X j if j 1,, P 1 Z j P 1 Z j j = 1, 2, 3, 4. P X 1 = X 1 X 1 2 X 1 = X 1Y 2 Z 6 P Z 1 = Z 2 Z 1 6 Z 1 = Z 2Z 6 P1 X 2 = X 1 X 3 X 2 = X 1 X 3 P1 Z2 = Z 3 Z 6 Z 2 = Z 3 Z 6 P X 3 = X 1 X 1 4 X 3 = X 1X 4 P Z 3 = Z 4 Z 1 6 Z 3 = Z 4Z 6 P1 X 4 = X 1 X 5 X 4 = X 1X 5 P1 Z 4 = Z 5 Z 6 Z 4 = Z 5Z 6. P 1 = 6 2 P x 1 L P = P1 L Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

34 Synthesizing g L = P L 1 Phase gate on the 1st logical qubit: (translate P L 1 P 1 ) Definition : P 1 X j P 1 {Ȳj if j = 1, X j if j 1,, P 1 Z j P 1 Z j j = 1, 2, 3, 4. P1 X 1 = X 1 X 2 X 1 = X 1Y 2 Z 6 P1 Z1 = Z 2 Z 6 Z 1 = Z 2 Z 6 P X 2 = X 1 X 1 3 X 2 = X 1X 3 P Z 2 = Z 3 Z 1 6 Z 2 = Z 3Z 6 P1 X 3 = X 1 X 4 X 3 = X 1X 4 P1 Z 3 = Z 4 Z 6 Z 3 = Z 4Z 6 P1 X 4 = X 1 X 5 X 4 = X 1X 5 P1 Z 4 = Z 5 Z 6 Z 4 = Z 5Z 6. P 1 = 6 2 P P x 1 L P = P1 L Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

35 Synthesizing g L = CZ L 12 Recollect: Definition : X 1 Z 2 if j = 1, CZ 12 X j CZ 12 Z 1 X 2 if j = 2,, X j if j 1, 2 CZ 12 Zj CZ 12 Z j j = 1, 2, 3, 4. 1 From rows of G X C/C, G Z C/C we get the logical Paulis: X 1 = X 1 X 2 = E(110000, ) Z1 = Z 2 Z 6 = E(000000, ) X 2 = X 1 X 3 = E(101000, ) Z2 = Z 3 Z 6 = E(000000, ) X 3 = X 1 X 4 = E(100100, ) Z 3 = Z 4 Z 6 = E(000000, ) X 4 = X 1 X 5 = E(100010, ) Z4 = Z 5 Z 6 = E(000000, ) 2 Cliff 2 6 = Sp(12, F 2 ): CZ 12 E(a, b)cz 12 = ±E ( [a, b]f CZ12 ). Find FCZ12. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

36 Finding CZ 12 via Sp(2m = 12, F 2 ) CZ X 1 = X 1 X 12 γ,φ 2 X1 X 2 Z 3 Z 6 [110000, ]F CZ12 = [110000, ] CZ X 2 = X 1 X 12 γ,φ 3 X1 X 3 Z 2 Z 6 [101000, ]F CZ12 = [101000, ] One possible solution [ ] I6 B F CZ12 =, B = I ( ) CZ 12 = diag ι vbv T Z 6 = CZ 36 CZ 26 CZ 23 Z Not captured in F CZ12 added to fix signs Z Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

37 Synthesizing g L = H L 1 Definition : H1 Xj H 1 { { Zj if j = 1, Xj X j if j 1,, H1 Zj H 1 if j = 1, Z j if j 1,. One solution: (note that F H 1 is not of any elementary symplectic form) [ ] A B F H 1 =, A = C D , B = , C = , D = Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

38 Decomposition of a Symplectic Matrix [Can17] (A, B) [ ] Q 0 A Q = [ ] 0 Q T Ik 0 R k I m k [ ] A B Symplectic Matrix F = C D I m k [ ] Im R T R2 = 2 with R 0 I 2 = [ ] m Ik Finally, A Q 1 1 (I m, 0) G k Ω [ ] Rk FA Q 1T R2 G k Ω = Ω T R1 Ω. 2 So F = A Q1 Ω T R1 G k T R2 A Q2. Translate each matrix to a Clifford element! Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

39 Synthesizing H L 1 for the [[6, 4, 2]] Code Our solution using the above general decomposition: H H H H H Particular solution for this case from [CR17]: H H H X Z Note: Both circuits correspond to the same symplectic solution F H1. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

40 Algorithm to Synthesize Logical Clifford Operators 1 Determine the target ḡ by specifying its action on X i, Z i [Got09]: ḡ X i ḡ = X i, ḡ Z i ḡ = Z i. Add conditions to normalize or centralize S. 2 Using the maps γ, φ, transform these relations into linear equations on Fḡ Sp(2m, F 2 ), i.e., ḡe(a, b)ḡ = ±E ([a, b]fḡ ) [a, b] [a, b]fḡ. 3 Find the feasible symplectic solution set Fḡ using transvections. 4 Factor each Fḡ Fḡ using above decomposition [Can17], and compute the physical Clifford operator ḡ. 5 Check for conjugation of ḡ with S, X i, Z i. If some signs are incorrect, post-multiply by an element from HW N as necessary to satisfy these conditions (apply [NC10, Prop. 10.4] to S = S, X i, Z i ). 6 Express ḡ as a sequence of physical Clifford gates obtained from the factorization in step 4. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

41 Overview 1 Mathematical Setup for Quantum Computing 2 Logical Pauli Operators for CSS Codes 3 Logical Clifford Operators for Stabilizer Codes 4 General Algorithms and Results Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

42 Symplectic Transvections Definition: Given a vector h F 2m 2, the transvection Z h : F 2m 2 F 2m 2 is Z h (x) x + x, h s h F h = I 2m + Ωh T h Sp(2m, F 2 ). Fact: Transvections generate the binary symplectic group Sp(2m, F 2 ). Lemma ([SAF08; KS14]) Let x, y F 2m 2. Then there exists at most two transvections F h 1, F h2 s.t. xf h1 F h2 = y. We extend this to map a sequence of vectors x i to y i, i = 1,..., t. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

43 Solving for Symplectic F s.t. x i F = y i, i = 1,..., t Input: x i, y i F 2m 2 s.t. x i, x j s = y i, y j s i, j {1,..., t}. Output: F Sp(2m, F 2 ) satisfying x i F = y i i {1,..., t} 1: if x 1, y 1 s = 1 then 2: set h 1 x 1 + y 1 and F 1 F h1. 3: else 4: h 11 w 1 + y 1, h 12 x 1 + w 1 and F 1 F h11 F h12. 5: end if 6: for i = 2,..., t do 7: Calculate x i x i F i 1 and x i, y i s. 8: if x i = y i then 9: Set F i F i 1. Continue. 10: end if 11: if x i, y i s = 1 then 12: Set h i x i + y i, F i F i 1 F hi. 13: else 14: Find a w i s.t. x i, w i s = y i, w i s = 1 and y j, w i s = y j, y i s j < i. 15: Set h i1 w i + y i, h i2 x i + w i, F i F i 1 F hi1 F hi2. 16: end if 17: end for 18: return F F t. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

44 Algorithm to Solve for all Symplectic Solutions Input: u a, v b F 2m 2 s.t. u a, v b s = δ ab and u a, u b s = v a, v b s = 0, a, b {1,..., m}. u i, v j F 2m 2 s.t. u i 1, u i 2 s = 0, v j 1, v j 2 s = 0, u i, v j s = δ ij, where i, i 1, i 2 I, j, j 1, j 2 J, I, J {1,..., m}. Output: F Sp(2m, F 2 ) s.t. each F F satisfies u i F = u i i I, & v j F = v j j J. 1: Determine a particular symplectic solution F 0 for the linear system. 2: Form the matrix A whose a-th row is u a F 0 and (m + b)-th row is v b F 0, where a, b {1,..., m}. 3: Compute the inverse of this matrix, A 1, in F 2. 4: Set F = φ and α Ī + J, where Ī, J denote the set complements of I, J in {1,..., m}, respectively. 5: for l = 1,..., 2 α(α+1)/2 do 6: Form a matrix B l = A. 7: For i / I and j / J replace the i-th and (m +j)-th rows of B l with arbitrary vectors such that B l ΩBl T = Ω and B l B l for 1 l < l. 8: Compute F = A 1 B. 9: Add F l F 0 F to F. 10: end for 11: return F Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

45 More Results... Given a stabilizer code with logical Paulis X i, Z i, we have the system γ( X ) γ( X ) γ(s) F = γ(s ). γ( Z) γ( Z ) Theorem For an [[m, m k]] stabilizer code, the number of symplectic solutions for each logical Clifford operator is 2 k(k+1)/2. Theorem For each logical Clifford operator of an [[m, m k]] stabilizer code, one can always synthesize a solution that centralizes the stabilizer S. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

46 Recap: In this talk... Synthesis of logical Pauli operators for CSS codes Two popular algorithms in the literature: [Got97b; Wil09]. We provided a closely-related but classical coding-theoretic perspective. Synthesis of logical Clifford operators for stabilizer codes Methods seem to exist only for particular QECCs and operations, e.g., [Got97a; Fow+12; GR13; CR17]. We proposed a systematic framework using symplectic geometry. For an [[m, m k]] stabilizer code, we showed that there are 2 k(k+1)/2 symplectic solutions for a given logical Clifford operator. Then enumerated these matrices efficiently using symplectic transvections, and translated them to physical operators (circuits). Showed that any normalizing solution can be converted into a centralizing solution, i.e., commute with every stabilizer element. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

47 Future Work How to leverage this efficient enumeration during the process of computation? Understand the geometry of the solution space of symplectic matrices. Optimization of solutions with respect to a useful metric. Decomposition of symplectic matrix motivated by practical constraints, e.g., circuit complexity, fault-tolerance. Extend the framework to accommodate non-clifford gates, e.g., T.... etc. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

48 References [CS96] [Ste96] [Got97a] [Got97b] A. R. Calderbank and Peter W. Shor. Good quantum error-correcting codes exist. In: Phys. Rev. A 54 (2 Aug. 1996), pp A. M. Steane. Simple quantum error-correcting codes. In: Phys. Rev. A 54.6 (1996), pp doi: /PhysRevA Daniel Gottesman. A Theory of Fault-Tolerant Quantum Computation. In: arxiv preprint arxiv:quant-ph/ (1997). Daniel Gottesman. Stabilizer codes and quantum error correction. PhD thesis. California Institute of Technology, Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

49 References [Cal+98] [DD03] [SAF08] [Got09] A.R. Calderbank et al. Quantum error correction via codes over GF(4). In: IEEE Trans. Inform. Theory 44.4 (July 1998), pp Jeroen Dehaene and Bart De Moor. Clifford group, stabilizer states, and linear and quadratic operations over GF(2). In: Phys. Rev. A 68.4 (Oct. 2003), p doi: /PhysRevA A Salam, E Al-Aidarous, and A El Farouk. Optimal symplectic Householder transformations for SR decomposition. In: Lin. Algebra and its Appl. 429 (2008), pp doi: /j.laa Daniel Gottesman. An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation. In: arxiv preprint arxiv: (2009). Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

50 References [Wil09] [NC10] [Fow+12] [GR13] Mark M Wilde. Logical operators of quantum codes. In: Phys. Rev. A 79.6 (2009), pp doi: /PhysRevA Michael A Nielsen and Isaac L Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Austin G. Fowler et al. Surface codes: Towards practical large-scale quantum computation. In: arxiv prepreint arxiv: (2012). [Online]. Available: Markus Grassl and Martin Roetteler. Leveraging automorphisms of quantum codes for fault-tolerant quantum computation. In: Proc. IEEE Int. Symp. Inform. Theory. IEEE, July 2013, pp Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

51 References [KMM13] [KS14] [Can17] [CR17] Vadym Kliuchnikov, Dmitri Maslov, and Michele Mosca. Asymptotically Optimal Approximation of Single Qubit Unitaries by Clifford and T Circuits Using a Constant Number of Ancillary Qubits. In: Phys. Rev. Lett (May 2013), p doi: /PhysRevLett Robert Koenig and John A Smolin. How to efficiently select an arbitrary Clifford group element. In: J. Math. Phys (Dec. 2014), p doi: / Trung Can. An Algorithm to Generate a Unitary Transformation from Logarithmically Many Random Bits. Research Independent Study Report, Duke Univeristy Rui Chao and Ben W Reichardt. Fault-tolerant quantum computation with few qubits. In: arxiv preprint arxiv: (2017). Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47

52 Thank you! Questions? For details see Have fun synthesizing Clifford circuits for your favorite stabilizer code, at :-). Any feedback is much appreciated. Narayanan Rengaswamy Logical Clifford Operators in Q. Computing arxiv: / 47