Stabilizer Codes for Prime Power Qudits Daniel Gottesman Perimeter - PowerPoint PPT Presentation

Stabilizer Codes for Prime Power Qudits Daniel Gottesman Perimeter Institute

Qubit Pauli and Clifford Groups # qubits register dimension P n,2 X b | v 〉 = | v+b 〉 Pauli group i a X b Z c = (-1) c · v | v 〉 Z c | v 〉 discard c(X b Z c , X b’ Z c’ ) = phase b · c’ - b’ · c Symplectic representation (Z 2 ) 2n ( b | c ) b , c ∈ (Z 2 ) n a ∈ Z 4 Clifford group C n,2 = {U | UPU † ∈ P n,2 } = ∑ y (-1) xy | y 〉 Hadamard H | x 〉 phase violates Phase P | x 〉 = i x | x 〉 binary arithmetic CNOT | x, y 〉 = | x, x+y 〉

Qubit Stabilizer Codes A qubit stabilizer S is an Abelian subgroup of P n,2 which does not contain -I. The code space corresponding to S is { | ψ 〉 | M | ψ 〉 = | ψ 〉 ∀ M ∈ S} Example: 5-qubit code [[5,1,3]] n physical qubits X Z Z X I r = n-k stabilizer generators M 1 , ..., M r I X Z Z X k logical qubits X I X Z Z Other elements of S are products Z X I X Z of generators. E.g.: Z Z X I X = M 1 M 2 M 3 M 4 for 5-qubit code Error syndrome: s (P) = {c(M 1 ,P), c(M 2 ,P), ..., c(M r ,P)} ∈ (Z 2 ) r E.g., for 5-qubit code, s (Y 3 ) = 1110

Prime Dimension Pauli and Clifford Each register has prime dimension p X b | v 〉 = | v+b 〉 P n,p Pauli group ω a X b Z c = ω c · v | v 〉 Z c | v 〉 discard ω = e 2 π i / p phase c(X b Z c , X b’ Z c’ ) = Symplectic representation (Z p ) 2n b · c’ - b’ · c ( b | c ) b , c ∈ (Z p ) n a ∈ Z p Clifford group C n,p = {U | UPU † ∈ P n,p } = ∑ y ω xy | y 〉 Fourier F | x 〉 phase uses mod = ω x(x-1) | x 〉 p arithmetic, just Phase P | x 〉 like everything CNOT | x, y 〉 = | x, x+y 〉 else

Prime Dimensional Stabilizers A qudit stabilizer S is an Abelian subgroup of P n,p which does not contain ω I. The code space corresponding to S is { | ψ 〉 | M | ψ 〉 = | ψ 〉 ∀ M ∈ S} Example: 5-qudit code [[5,1,3]] p Z Z -1 X -1 n physical qudits X I r = n-k stabilizer generators M 1 , ..., M r Z Z -1 X -1 I X k logical qudits X -1 I X Z Z -1 Other elements of S are products of Z -1 X -1 I X Z generators, including powers 1, ..., p-1 E.g.: Z Z -1 X -1 I X = M 1-1 M 2-1 M 3-1 M 4-1 Error syndrome: s (P) = {c(M 1 ,P), c(M 2 ,P), ..., c(M r ,P)} ∈ (Z p ) r E.g., for 5-qubit code, s (X 3 Z 3 ) = (1,-1,1,0)

Composite Dimension For composite qudit dimension q, we can X b | v 〉 = | v+b 〉 do this too, using the same Pauli group = ω c · v | v 〉 Z c | v 〉 (often known as the Heisenberg-Weyl group). ω = e 2 π i / q c(X b Z c , X b’ Z c’ ) = This is workable, but the stabilizer b · c’ - b’ · c codes derived this way lack some of b , c ∈ (Z q ) n the standard structure of stabilizer a ∈ Z q codes for prime-dimensional qudits. For instance, not all elements of P n,q are equivalent (some have different orders), and there is no simple relationship between the number of generators of S and the number of logical qudits. There also do not need to be an integral number of qudits. When q=p m , it is better to use an alternate Pauli group based on the finite field of size q.

Finite Fields A field has Abelian addition and multiplication rules, including 0, 1, additive and multiplicative inverses, and a distributive law. Familiar examples of infinite fields are rationals, reals, & complex #s. The simplest finite fields are Z p , mod p arithmetic for prime p. Example: For any q = p m , there exists a unique GF(9) = Z 3 ( α ), finite field GF(q) of size q. Such a α 2 + α + 2 = 0 field can be constructed by taking Z p and adjoining the roots of irreducible Elements are 0, 1, 2, α , α polynomials. +1, α +2, 2 α , 2 α +1, 2 α +2 GF(q) has characteristic p, meaning E.g., α (2 α +1) = 2 α 2 + α any element added p times gives 0. = 2(- α -2) + α = 2 α +2

Z p Versus GF(p m ) GF(q), q=p m can be viewed as a vector space over Z p : pick m independent adjoining elements α 1 , ..., α m . Then the elements of GF(q) can all be written in the form ∑ i c i α i , with c i ∈ Z p . GF(q) = (Z p ) m The trace can be used to reduce elements of GF(q) to Tr elements of Z p : Z p m-1 2 tr x = x + x p + x p + ... + x p Properties of trace: 1. tr α ∈ Z p 2. tr ( α + β ) = tr α + tr β 3. tr ( α p ) = tr α 4. tr (a β ) = a tr β (for a ∈ Z p )

“Standard” Pauli Group for q=p m P n,q = { ω c X α Z β } X α | γ 〉 = | γ + α 〉 Z β | γ 〉 = ω tr β · γ | γ 〉 α , β ∈ GF(q) n , c ∈ Z p For qudits of dimension q=p m , the current preferred definition of the Pauli group takes advantage of the trace to allow the exponents of X and Z to be elements of GF(q), but the phase is still drawn from Z p . Commutation can also be determined via tr: c(X α Z β , X α ’ Z β ’ ) = tr α · β ’ - α ’ · β However, this definition of P n,q is isomorphic to P mn,p . That is, we actually have a p-dimensional Pauli group: Given basis { α 1 , ... , α m } for GF(q) over Z p , choose a dual basis { β 1 , ... , β m } with the property tr ( α i β j ) = δ ij . Then let α = ∑ i a i α i and β = ∑ j b j β j , so we can interpret X α = X a1 ⊗ X a2 ⊗ ... ⊗ X am q-dim. qudit broken up into m p-dim qudits Z β = Z b1 ⊗ Z b2 ⊗ ... ⊗ Z bm

“Standard” Stabilizers for q=p m Consequently, if stabilizers are defined in the usual way from this Pauli group P n,q , they are equivalent to mn-qudit stabilizers for p- dimensional qudits. Example: 5-qudit code [[5,1,3]] 9 Z -1 X -1 X Z I n physical qudits X α Z α Z - α X - α r stabilizer generators M 1 , ..., M r I k = n-r/m logical qudits Z -1 X -1 I X Z Other elements of S are products of X α Z α Z - α X - α I generators, including powers 1, ..., p-1. X -1 I X Z Z -1 Powers of α (for GF(9)) require X α Z α Z - α I additional generators. X - α Z -1 X -1 I X Z Z - α X - α X α Z α Error syndrome still a Z p vector I Error syndrome: s (P) = {c(M 1 ,P), c(M 2 ,P), ..., c(M r ,P)} ∈ (Z p ) r

True GF(q) Stabilizer Codes Note the example 5-qudit code has an extra symmetry as do most other interesting GF(q) stabilizer codes. In the symplectic representation, it is GF(q)-linear, not just Z p -linear: 1 0 0 -1 0 0 1 -1 0 0 0 0 0 0 0 0 α - α α - α 0 1 0 0 -1 0 0 1 -1 0 0 α 0 0 - α 0 0 α - α 0 -1 0 1 0 0 0 0 0 1 -1 - α 0 α 0 0 0 0 0 α - α 0 -1 0 1 0 -1 0 0 0 1 0 - α 0 α 0 - α 0 0 0 α However, since each generator can have an independent phase, so there is no clear meaning of the “multiplication by α ” symmetry in the Pauli group P n,q . It should mean “exponentiation by α ” but that is not a well-defined operation.

Lifted Pauli Group (Odd q) We want to lift the Pauli group to a larger group where exponentiation by elements of GF(q) is well-defined. We expand the set of possible phases to be all elements of GF(q): Ṗ n,q = { ω μ X α Z β } α , β ∈ GF(q) n , μ ∈ GF(q) ( ω μ X α Z β )( ω μ ’ X α ’ Z β ’ ) = ω μ + μ ’- α ’ · β X α + α ’ Z β + β ’ c(X α Z β , X α ’ Z β ’ ) = α · β ’ - α ’ · β ∈ GF(q) We can project an element of the lifted Pauli group back to the regular Pauli group by using tr on the phase: Ṗ n,q ω μ X α Z β 𝚸 (PQ) = ( 𝚸 P)( 𝚸 Q) 𝚸 c( 𝚸 P , 𝚸 Q) = tr c(P ,Q) 𝚸 P n,q ω tr μ X α Z β

Exponentiation (Odd q) phase and existing exponents get multiplied by γ ( ω μ X α Z β ) γ = ω γμ -[ γ ( γ -1)/2] α · β X γ α Z γ β new phase term giving phase Note that this formula reduces accumulation from to the correct one for γ ∈ Z p . “reorganizing” X and Z powers Exponentiation satisfies other standard properties: Because of the 1/2 that 1. P γ P δ = P γ + δ appears in the definition 2. (P γ ) δ = P γδ of exponentiation, this 3. P γ Q γ = (PQ) γ when c(P ,Q)=0 only works for odd q.

Pauli Group Vs. Lifted Pauli Group Exponentiation in Ṗ n,q lets us group together operators in P n,q whose symplectic representations are related by GF(q) multiplication: Example: Ṗ 1,9 P 1,9 P = ω 0 X 1 Z 1 ω 0 X 1 Z 1 𝚸 P 2 = ω 2 X 2 Z 2 ω 1 X 2 Z 2 P α = ω 1+ α X α Z α ω 2 X α Z α . . . . . . This single element is enough to generate all of the others, which correspond to m independent elements of P n,q . The single phase ω μ ( μ ∈ GF(q)) gives the m independent phases ω a (a ∈ Z p ). There is a unique correspondence P ∈ Ṗ n,q to { 𝚸 P γ } ⊂ P n,q .

Lifted Stabilizers S is a lifted stabilizer if S is an Abelian subgroup of Ṗ n,q closed under exponentiation (i.e., P ∈ S ⇒ P γ ∈ S ∀ γ ∈ GF(q)), with ω μ ∉ S. Thm.: The lifted stabilizers are in one-to-one correspondence with the true GF(q) stabilizers. 𝚸 S S Generalized eigenvalues: | ψ 〉 is a generalized eigenvector of P ∈ Ṗ n,q if it is an eigenvector of P γ ∀ γ ∈ GF(q). If it has eigenvalue ω ai for P γ i , then the generalized eigenvalue is ω μ s.t. tr( γ i μ ) = a i for all i. The codewords are the generalized ω 0 eigenvectors of the elements of the lifted stabilizer, and an error E alters the generalized eigenvalues, so the error syndrome is the GF(q) vector of generalized eigenvalues after E, given by c(E, M i ) for generators M i of the lifted stabilizer.