Introduction to quantum information processing

Transcription

1 Introduction to quantum information processing Stabilizer codes Brad Lackey 15 November 2016 STABILIZER CODES 1 of 17

2 OUTLINE 1 Stabilizer groups and stabilizer codes 2 Syndromes 3 The Normalizer STABILIZER CODES 2 of 17

3 LAST TIME... The Steane code as a CSS-code has generators: G 1 = and G 2 = G 2 is also the parity check matrix for G 1, so G 2 is used to compute the all the syndomes.. STABILIZER CODES 3 of 17

4 OUTLINE 1 Stabilizer groups and stabilizer codes 2 Syndromes 3 The Normalizer STABILIZER CODES 4 of 17

5 The Pauli group is defined to be: THE PAULI GROUP P n = {cσ 1 σ n : σ j {1, σ x, σ y, σ z }, c {±1, ±i}}. From σ x σ y = iσ z we have Lemma Two elements of P n either commute or anticommute. Definition Let S be an Abelian subgroup of P n \ { 1}. The stabilizer code of S is C(S) = { ψ : M ψ = ψ for all M S}. Basic idea: S is the set of syndrome operators for the code. 1 does not have a +1 eigenvalues and so is not useful for syndromes. STABILIZER CODES 5 of 17

6 CHECK ROWS To any element M = ασ 1 σ n we associate a row (x 1 x 2 x n z 1 z 2 z n ) F n 2 where the x and z bits are computed as x j = z j = { 1 if σj = σ x or σ y 0 if σ j = 1 or σ z { 1 if σj = σ y or σ z 0 if σ j = 1 or σ x. This is called the check row and denoted c(m). The coefficient α is ignored! Lemma c(m 1 M 2 ) = c(m 1 ) + c(m 2 ). Consequently: generators of S correspond to (linear) bases of c(s). We can use Gaussian elimination to find generating sets. STABILIZER CODES 6 of 17

7 EXAMPLE: THE STEANE CODE In the Steane code, both X-checks and Z-checks are computed by: G 2 = S = X 1 X 3 X 5 X 7, X 2 X 3 X 6 X 7, X 4 X 5 X 6 X 7, Z 1 Z 3 Z 5 Z 7, Z 2 Z 3 Z 6 Z 7, Z 4 Z 5 Z 6 Z 7. Here mean generated by. In fact, S has 64 elements. The check rows of S are the rows of the matrix: STABILIZER CODES 7 of 17

8 THE NORMALIZER Definition Let S be stabilizer group. Its normalizer is N(S) = {P P n : MP = PM for all M S}. This is usually called the centralizer. But these are the same: The normalizer is usually P P n such that P 1 MP S for all M S. Since elements of P n either commute or anticommute, we have P 1 MP = ±P 1 PM = ±M. So if both M, P 1 MP S then since M S we must have MP = PM. The normalizer plays two important roles for stabilizer codes: it defines logical Pauli operators on the code, and it characterizes uncorrectable Pauli errors. STABILIZER CODES 8 of 17

9 OUTLINE 1 Stabilizer groups and stabilizer codes 2 Syndromes 3 The Normalizer STABILIZER CODES 9 of 17

10 PAULI ERRORS From Assignment 5, #3, we need only focus on correction Pauli errors, E. Claim: for ψ C(S), we have E ψ is eigenvector of any M S. Proof: M(E ψ ) = ±E(M ψ ) = ±(E ψ ). Recall: the eigenvalue is the syndrome, so we get syndromes directly. Let {M α } n k α=1 be a generating set for a stabilizer S. The syndrome vector of E P n is s(e) = (s α (E)) {±1} n k, where it can be computed from EM α = s α (E) M α E. Syndromes satisfy: s α (E 1 E 2 ) = s α (E 1 ) s α (E 2 ). The eigenprojection for a general vector v = (v α ) {±1} n k is Π v = 1 n k 2 n k (1 v α M α ). α=1 The normalizer N(S) is the set of Pauli operators with all +1 syndromes. STABILIZER CODES 10 of 17

11 Lemma SYNDROME MEASUREMENTS We have {Π v } v {±1} n k is a P.O.V.M. Proof : We have v {±1} n k (1 + v α M α ) = n k α=1 v α=±1 (1 + v α M α ) = α A 2 1 = 2 n k 1. Therefore, v {±1} n k Π v {±1} n k = 1 2 n k 2 n k 1. For a general vector v = (v α ) {±1} n k define the syndrome space: C v = image(π v ). (Note: C(S) = C (+1,,+1).) STABILIZER CODES 11 of 17

12 SYNDROME SPACES AND ERROR RECOVERY Theorem For a Pauli error E, we have E : C(S) C s(e) unitarily. Proof : We claim EΠ C(S) = Π s(e) E, which follows from: 1 2 n k E α ( ) (1 + M α ) = 1 2 n k (1 + s α (E)M α ) E. α Thus E ψ = Π s(e) E ψ for ψ C(S), and so E ψ C s(e). Here s the error correction process: 1 Begin with ψ C(S). Error results in E ψ. 2 Measure {Π v } v {±1} n k. 3 The result is v = s(e) with certainty as E ψ C s(e). 4 After measurement state is still E ψ. Apply E. STABILIZER CODES 12 of 17

13 OUTLINE 1 Stabilizer groups and stabilizer codes 2 Syndromes 3 The Normalizer STABILIZER CODES 13 of 17

14 STABILIZERS VERSUS NORMALIZERS Let S be a stabilizer group. Recall: The normalizer is N(S) = {P P n : MP = PM for all M S}. Since S is Abelian (all elements commute), S N(S). The syndrome of any E N(S) is (+1,..., +1). Therefore, any E N(S) maps E : C(S) C(S) What does this mean for error correction? If E S, then E ψ = ψ for all ψ C(S) (so no error). It turns out, if E ψ = ψ for all ψ C(S), then E S. We won t prove this because we didn t developed the needed machinery: one characterizes S and N(S) using augmented check matrices. Consequently, if E N(S) \ S then E ψ ψ for some ψ C(S). So if E is unintentional: this is an error. If E is intentional: we performed an quantum operation on our encoded data. More on this topic next time. STABILIZER CODES 14 of 17

15 QUANTUM ERROR CORRECTION CONDITION Theorem Let S be a stabilizer group and N(S) its normalizer. The stabilizer code C(S) can correct Pauli errors {E j } r j=1 if for each j k we have E j E k N(S) \ S. Proof : If E j E k S then Π C(S) E j E k Π C(S) = Π C(S). If E j E k N(S) then it anticommutes with some generator M α of S. So, E j E k Π C(S) = 1 2 n k E j E k (1+M β ) = 1 2 n k (1 M α)e j E k (1+M β ). β β α But, Π(1 M α ) = 1 2 n k (1 + M β )(1 + M α )(1 M α ) = 0. β α Thus, Π C(S) E j E k Π C(S) = 0. From the QECC we can correct {E j } r j=1. STABILIZER CODES 15 of 17

16 Corollary DISTANCE If every element in N(S) \ S involves at least d Pauli operators, then C(S) can correct t = d 1 2 errors. Proof : If any E j has at most t Pauli operators, then E j E k has at most 2t d. Therefore, E j E k N(S) \ S. The d in this results is called the distance of the stabilizer code. This allows one to find how many errors a quantum code can correct. However, if possible, it s easier to show each error has a unique syndrome: if s(e j ) s(e k ) then some α has s α (E j E k) = s α (E j ) s α (E k ) = 1, hence E j E k N(S). So the theorem shows we can correct {E j }. STABILIZER CODES 16 of 17

17 NEXT TIME... Fault-tolerate error-correction and quantum computation. STABILIZER CODES 17 of 17