Stabilizer quantum codes via the CWS framework Leonid Pryadko - - PowerPoint PPT Presentation
Stabilizer quantum codes via the CWS framework Leonid Pryadko University of California, Riverside Y. Li, I. Dumer, & LPP, PRL (2010) Y. Li, I. Dumer, M. Grassl, & LPP, PRA (2010) A. Kovalev, I. Dumer, & LPP, PRA (accepted) NSF ARO
1-1
Leonid Pryadko
University of California, Riverside
1-2
Leonid Pryadko
University of California, Riverside
bound for codes from a given graph
2-1
Stabilizer code Q is determined by an Abelian stabilizer group S of Pauli operators Q ≡ {|ψ : S |ψ = |ψ , ∀S ∈ S } If S = G1, . . . , Gn−k, with (n − k) generators, the code encodes k logical qubits. There are k logical operators Xi, Zi, i = 1, . . . , k which commute with every element in S . The code is denoted [[n, k, d]], where d is the distance of the code. The group G1, . . . , Gn−k, Z1, . . . , Zk stabilizes a unique stabilizer state |s ≡ |¯ 0 . . . ¯ 0; the basis of the code is |α1, . . . αk ≡ X
α1 1 . . . X αk k |s, αj = {0, 1}, j = 1, . . . , k.
Errors are detected by measuring the generators Gi of the stabilizer S General quantum code is a subspace Q of n-qubit Hilbert space H⊗n
2 .
3-1
Q ≡ {|ψ : Gi |ψ = |ψ , i = 1, . . . , 4} with generators G1 = XZZXI, G2 = IXZZX, G3 = XIXZZ, G4 = ZXIXZ A basis of the code space is (up to normalization) |¯ 0 =
4
(1 + Gi) |00000 , |¯ 1 = X |¯ 0 . The logical operators can be taken as X = ZZZZZ, Z = XXXXX. Measure generators of the stabilizer to find the error, e.g.,
ψ
0 + B |¯ 1) gives unique syndrome G1 = 1, G2 = 1, G3 = 1, G4 = −1. For this code, there are total of 15 single-qubit errors, and exactly 15 distinct syndromes (apart from Gi = 1 for any |ψ ∈ Q).
4-1
For a simple graph G = (V, E) with adjacency matrix R ≡ γij, the generators Si ≡ Xi
n
Zγij These define the Abelian graph stabilizer group SG ≡ S1, . . . , Sn and the graph state |s: Si |s = |s, a [[n, 0, d]] stabilizer code Distance of a graph state is defined as the minimum weight of an element of the graph stabilizer SG.
4-2
For a simple graph G = (V, E) with adjacency matrix R ≡ γij, the generators Si ≡ Xi
n
Zγij These define the Abelian graph stabilizer group SG ≡ S1, . . . , Sn and the graph state |s: Si |s = |s, a [[n, 0, d]] stabilizer code Example: Ring graph for n = 3; S1 = XZZ, S2 = ZXZ, S3 = ZZX. |s is an equal superposition of all 23 states, taken with positive or negative signs depending on the number of pairs of ones at positions connected by the edges of the graph.
|s = |000 + |001 + |010 − |011 + |100 − |101 − |110 − |111 = S2|s = |010 − |011 + |000 + |001 − |110 − |111 + |100 − |101
Distance of a graph state is defined as the minimum weight of an element of the graph stabilizer SG.
5-1
Generally non-additive, but include all stabilizer codes as a subclass. Standard form: Q = (G, C). Graph G → graph state |s Classical binary code (n, K, d) = C = {ci}K
i=1, with n-bit ci.
Quantum basis vectors |i ≡ Wi |s codeword operators Wi ≡ Zci,1 . . . Zci,n. Error E = ZuXv maps to binary vector [ClG(E)]j ≡ uj +γijvi Invented by Cross, Smith, Smolin & Zeng (2007). Example: Non-additive CWS code ((5, 6, 2)). The n = 5 ring graph generated by S2 = ZXZII and cyclic permutations. Classical codewords c0 = 00000, c1 = 01101, c2 = 10110, c3 = 01011, c4 = 10101, c5 = 11010.
5-2
Generally non-additive, but include all stabilizer codes as a subclass. Standard form: Q = (G, C). Graph G → graph state |s Classical binary code (n, K, d) = C = {ci}K
i=1, with n-bit ci.
Quantum basis vectors |i ≡ Wi |s codeword operators Wi ≡ Zci,1 . . . Zci,n. Error E = ZuXv maps to binary vector [ClG(E)]j ≡ uj +γijvi Invented by Cross, Smith, Smolin & Zeng (2007). Example: Non-additive CWS code ((5, 6, 2)). The n = 5 ring graph generated by S2 = ZXZII and cyclic permutations. Classical codewords c0 = 00000, c1 = 01101, c2 = 10110, c3 = 01011, c4 = 10101, c5 = 11010. Unfortunately, no known efficient algorithm to decode non-additive CWS codes.
6-1
Error mapping to binary vector [ClG(E)]j ≡ uj + γijvi Error detection condition i| E |j = CEδij (a) Non-degenerate case CE = 0: 0 = i| E |j = s| W †
i EWjS |s = ± s| W † i Wj(ES) |s.
If E = XvZu, get rid of all X operators with Si: vi = 0 Power of Z: ci ⊕ cj ⊕ ClG(E) If this is non-zero, classical and quantum error detection conditions coinside (b) Degenerate case CE = 0. Nothing to do in classical case. Quantum: E must commute with every Wi ⇒ ci, v = 0
7-1
not exceed that of the graph state induced by G [Grassl et al., 2009], dnon−deg
Q
≤ d′
G
the distance of the CWS code Q = (G, C) does not exceed the weight of Sj, dQ ≤ wgt Sj.
7-2
not exceed that of the graph state induced by G [Grassl et al., 2009], dnon−deg
Q
≤ d′
G
the distance of the CWS code Q = (G, C) does not exceed the weight of Sj, dQ ≤ wgt Sj.
distance of a CWS code Q = (G, C) cannot exceed r + 1. Consequences
code C involves all bits cannot be bigger than the minimum weight of Si.
8-1
nary code C = (11111) and the 5-ring graph G. Graph stabilizer generators S2 = Z1X2Z3 and cyclic shifts. Stabilizer gener- ators S2S3 = Z1Y2Y3Z4 and cyclic shifts. 4 3 2 1 5
8-2
nary code C = (11111) and the 5-ring graph G. Graph stabilizer generators S2 = Z1X2Z3 and cyclic shifts. Stabilizer gener- ators S2S3 = Z1Y2Y3Z4 and cyclic shifts.
the binary code C = (011100) and the graph shown. Code degenerate above the graph distance d′(G) = 2: S1S2 = X1X2. 1 2 3 5 6 4 3 2 1 5
8-3
nary code C = (11111) and the 5-ring graph G. Graph stabilizer generators S2 = Z1X2Z3 and cyclic shifts. Stabilizer gener- ators S2S3 = Z1Y2Y3Z4 and cyclic shifts.
the binary code C = (011100) and the graph shown. Code degenerate above the graph distance d′(G) = 2: S1S2 = X1X2. 1 2 3 5 6
generators X1X2X3X4, Z1Z2Z3Z4 and their cyclic shifts. It is generated by the code C = (1110000) and the graph shown. No graph with explicit circulant symmetry ex- ists. 1 2 3 4 5 6 7 4 3 2 1 5
9-1
Background: GF(4) map for the stabilizer of an additive code: U ≡ im′ZuXv → (v, u) → u + ωv, where ω is the generator of GF(4): ω ≡ ω2 = ω + 1, ωω = 1. Operators U1 and U2 commute iff e1 ∗ e2 ≡ e1 · e2 + e1 · e2 = v1 · u2 + u1 · v2.
Parity check matrix for the binary code Graph adjacency matrix
9-2
Background: GF(4) map for the stabilizer of an additive code: U ≡ im′ZuXv → (v, u) → u + ωv, where ω is the generator of GF(4): ω ≡ ω2 = ω + 1, ωω = 1. Operators U1 and U2 commute iff e1 ∗ e2 ≡ e1 · e2 + e1 · e2 = v1 · u2 + u1 · v2.
Parity check matrix for the binary code Graph adjacency matrix G is automatically self-orthogonal as long as R = Rt G ∗ G
t = P(ω1 + R) ∗ (ω1 + R)P t = 0
10-1
Theorem: For a given graph G with the graph-state distance d′(G), there is a binary code C such that the CWS code Q = (G, C) is pure and satisfies the quantum Gilbert-Varshamov bound,
d−1
3s n s
10-2
Theorem: For a given graph G with the graph-state distance d′(G), there is a binary code C such that the CWS code Q = (G, C) is pure and satisfies the quantum Gilbert-Varshamov bound,
d−1
3s n s
code – much easier problem When n, k, d → ∞, relative distance δ = d/n and code rate R = k/n satisfy δ log2 3 + H2(δ) ≤ 1 − R, H2(δ) ≡ −δ log2 δ−(1−δ) log2(1−δ)
11-1
Gilbert-Varshamov bound for finite distance d implies the redundancy n − k = d log2(3n/2d). Example: Codes on finite square lattice have distance d ≤ 5. With edges involved in the code, d ≤ 4. Code [[25, 4, 4]], member of the family [[LxLy, (Lx − 3)(Ly − 3), 4]]. Redundancy n−k = 3(Lx+Ly)−9 ∝ √n. Codes found numerically: [[25, 10, 5]], [[25, 13, 4]], [[25, 17, 3]]
11-2
Gilbert-Varshamov bound for finite distance d implies the redundancy n − k = d log2(3n/2d). Example: Codes on finite square lattice have distance d ≤ 5. With edges involved in the code, d ≤ 4. Code [[25, 4, 4]], member of the family [[LxLy, (Lx − 3)(Ly − 3), 4]]. Redundancy n−k = 3(Lx+Ly)−9 ∝ √n. Codes found numerically: [[25, 10, 5]], [[25, 13, 4]], [[25, 17, 3]] Generalization: codes on a D-dimensional hypercubic lattice with n = LD, d = 2D, and redundancy n − k ∝ LD−1.
12-1
Cyclic code: (ZY Y ZI) ∈ S → (IZY Y Z) ∈ S . Circulant matrix G = 1 ω ω 1 · · 1 ω ω 1 1 · 1 ω ω ω 1 · 1 ω ω ω 1 · 1 Map: circulant matrix → polynomial g(x) = 1+ ¯ ωx+ ¯ ωx2 +x3 Shift: g(x) → xg(x) mod (xn − 1)
12-2
Cyclic code: (ZY Y ZI) ∈ S → (IZY Y Z) ∈ S . Circulant matrix G = 1 ω ω 1 · · 1 ω ω 1 1 · 1 ω ω ω 1 · 1 ω ω ω 1 · 1 Map: circulant matrix → polynomial g(x) = 1+ ¯ ωx+ ¯ ωx2 +x3 Shift: g(x) → xg(x) mod (xn − 1) CWS cyclic code: g(x) = p(x)(ω + r(x)), where r(x) is symmetric, r(xn−1) = r(x) mod (xn − 1), and p(x) is a factor of xn − 1, p(x)q(x) = xn − 1. Example: [[5, 1, 3]] code p(x) = 1 + x, r(x) = x + x4. Additive CWS code: G = P(ω1 + R), Rt = R.
12-3
Cyclic code: (ZY Y ZI) ∈ S → (IZY Y Z) ∈ S . Circulant matrix G = 1 ω ω 1 · · 1 ω ω 1 1 · 1 ω ω ω 1 · 1 ω ω ω 1 · 1 Map: circulant matrix → polynomial g(x) = 1+ ¯ ωx+ ¯ ωx2 +x3 Shift: g(x) → xg(x) mod (xn − 1) CWS cyclic code: g(x) = p(x)(ω + r(x)), where r(x) is symmetric, r(xn−1) = r(x) mod (xn − 1), and p(x) is a factor of xn − 1, p(x)q(x) = xn − 1. Example: [[5, 1, 3]] code p(x) = 1 + x, r(x) = x + x4. Additive CWS code: G = P(ω1 + R), Rt = R. General one-generator cyclic codes p(x)p(xn−1)r(xn−1) = p(x)p(xn−1)r(x) mod xn − 1.
13-1
Consider a binary cyclic code C[n, k, dC] with the generator polynomial q(x) which is irreducible. Then there exists a quantum cyclic code Q[n, k, d], with d = min(dC, d′), where d′ = dGV(n, k) is (a somewhat improved) Gilbert-Varshamov bound if q(x) is non-palindromic, dGV(n, k) ≡ max d :
d−1
(3s − 3)gcd(s, n) n n s
while for a palindromic q(x), xdeg q(x)q(1/x) = q(x), d′ ≈ dGV(n/2, k/2) ≥ 1 2dGV(n, k).
14-1
Standard Gilbert-Varshamov bound for a finite number of encoded qubits k implies a finite relative distance d/n ≥ 0.189, n → ∞ Toric codes: [[n = 2L2, k = 2, d = L]]: d/n ∝ 1/√n [[18, 2, 3]], [[50, 2, 5]], [[98, 2, 7]], . . .
14-2
Standard Gilbert-Varshamov bound for a finite number of encoded qubits k implies a finite relative distance d/n ≥ 0.189, n → ∞ Toric codes: [[n = 2L2, k = 2, d = L]]: d/n ∝ 1/√n [[18, 2, 3]], [[50, 2, 5]], [[98, 2, 7]], . . . For n 50, most “good” quantum cyclic codes have dk = n, including many codes with small-weight stabilizer generators. Obtained from k replicas of the classical repetition codes [kd, k, d].
14-3
Standard Gilbert-Varshamov bound for a finite number of encoded qubits k implies a finite relative distance d/n ≥ 0.189, n → ∞ Toric codes: [[n = 2L2, k = 2, d = L]]: d/n ∝ 1/√n [[18, 2, 3]], [[50, 2, 5]], [[98, 2, 7]], . . . For n 50, most “good” quantum cyclic codes have dk = n, including many codes with small-weight stabilizer generators. Obtained from k replicas of the classical repetition codes [kd, k, d]. Codes corresponding to binary BCH codes: [[23, 12, 4]], [[47, 24, d ≥ 6]], [[71, 36, d ≥ 7]], [[95, 59, 5]], [[115, 71, 5]], [[143, 83, 11]], . . .
15-1
Take p(x) = 1 + x and n = t2 + (t + 1)2 = t + 2 + (t + 1)(2t − 1), t = 1, 2, . . . (n = 5, 13, 25, 41, . . .) ⇒ two code families [[n, 1, d]] with d = 2t + 1: ZY IY Z, ZY IIIY Z, ZY IIIIIY Z, . . . ZY Y Z, ZIY Y IZ, ZIIY Y IIZ, . . . Largest distance codes with weight-4 generators. [[13, 1, 5]] ZIY Y IZ 10ωω01
16-1
G → Q = (G, C). – Lower compexity compared to full search; Ok codes. – Use regular lattices to build finite-distance codes.
– Large class of easy-to construct quantum codes – Include codes with small-weight stabilizer – Rotated surface codes & friends – see Poster. – [[5, 1, 3]] code gets a torus.