. High-precision threshold of the toric code from spin-glass theory and graph polynomials . . . Masayuki Ohzeki Kyoto University 2015/07/01 This is in collaboration with Prof. Jesper L. Jacobsen (ENS). J. Phys. A: Math. Theor. 48 095001 (2015) [IOP select] Supported by the JSPS core-to-core program, and MEXT KAKENHI (No.15H03699) . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 1 / 15
| Ψ > | Φ > . Toric code . . Set a physical qubit on each edge of the square lattice on a torus. The stabilizer operators are ∏ ∏ σ z σ x Z p = X s = ( ij ) . ( ij ) ( ij ) ∈ ∂ p ( ij ) ∈ ∂ s They are commutable and the stabilizer state satisfies Z p | Ψ � = | Ψ � ( ∀ p ) X s | Ψ � = | Ψ � ( ∀ s ) . . . Z p X S The block denotes the physical qubit. . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 2 / 15
| Ψ > . Trivial cycle = Stabilizer operators . . The stabilizer state can be characterized by a product of the operators | Ψ( V ∗ , V ) � = ∏ ∏ X s | Φ � (1) Z p p ∈ V ∗ s ∈ V We use the degeneracy as redundancy of the logical qubits. ∑ | Ψ( V ∗ , V ) � | Ψ 0 � ∝ V ∗ , V . . . Z p Z p Z p Z p Z p . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 3 / 15
| Ψ > | Ψ > . Nontrivial cycle = Logical operators . . Let us introduce the “logical” operators ∏ ∏ σ z σ x Z h = X v = ( ij ) , ( ij ) ( ij ) ∈ L h ( ij ) ∈ L v and X h and Z v . ( Z h and X v , which commutes with each other, Z p and X s ) . . . . Encode . . We have four (2 2 ) different logical states. ∑ | Ψ( V ∗ , V ) � . | Ψ Z h � ∝ Z h X V X V V ∗ , V . . . . Z h Computation . . We can implement the Pauli operator Z h { X h , Z v } = 0 on the toric code. . . . . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 4 / 15
| Ψ > | Ψ > . Nontrivial cycle = Logical operators . . Let us introduce the “logical” operators ∏ ∏ σ z σ x Z h = X v = ( ij ) , ( ij ) ( ij ) ∈ L h ( ij ) ∈ L v and X h and Z v . ( Z h and X v , which commutes with each other, Z p and X s ) . . . . Encode . . We have four (2 2 ) different logical states. ∑ | Ψ( V ∗ , V ) � . | Ψ Z h � ∝ Z h X V X V V ∗ , V . . . . Z h Computation . . We can implement the Pauli operator Z h { X h , Z v } = 0 on the toric code. . . . . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 4 / 15
| Ψ > | Ψ > . Nontrivial cycle = Logical operators . . Let us introduce the “logical” operators ∏ ∏ σ z σ x Z h = X v = ( ij ) , ( ij ) ( ij ) ∈ L h ( ij ) ∈ L v and X h and Z v . ( Z h and X v , which commutes with each other, Z p and X s ) . . . . Encode . . We have four (2 2 ) different logical states. ∑ | Ψ( V ∗ , V ) � . | Ψ Z h � ∝ Z h X V X V V ∗ , V . . . . Z h Computation . . We can implement the Pauli operator Z h { X h , Z v } = 0 on the toric code. . . . . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 4 / 15
. Error model . . The error chain (flip ( σ x ( ij ) ) and phase ( σ z ( ij ) ) errors) appears following ( e 2 K p = 1 − p ) P ( E ) = p | E | (1 − p ) N B −| E | ∝ e K p τ E ∏ ij p ij where τ E ij = 1 for ij ∈ E and τ E ij = − 1 for ij / ∈ E . . . p . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 5 / 15
. Error model . . The error chain (flip ( σ x ( ij ) ) and phase ( σ z ( ij ) ) errors) appears following ( e 2 K p = 1 − p ) P ( E ) = p | E | (1 − p ) N B −| E | ∝ e K p τ E ∏ ij p ij where τ E ij = 1 for ij ∈ E and τ E ij = − 1 for ij / ∈ E . . . p . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 5 / 15
. Error model . . The error chain (flip ( σ x ( ij ) ) and phase ( σ z ( ij ) ) errors) appears following ( e 2 K p = 1 − p ) P ( E ) = p | E | (1 − p ) N B −| E | ∝ e K p τ E ∏ ij p ij where τ E ij = 1 for ij ∈ E and τ E ij = − 1 for ij / ∈ E . . . p Error correction strategy Connection between two ends of error chains . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 5 / 15
. Error model . . The error chain (flip ( σ x ( ij ) ) and phase ( σ z ( ij ) ) errors) appears following ( e 2 K p = 1 − p ) P ( E ) = p | E | (1 − p ) N B −| E | ∝ e K p τ E ∏ ij p ij where τ E ij = 1 for ij ∈ E and τ E ij = − 1 for ij / ∈ E . . . p Error correction strategy Connection between two ends of error chains . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 5 / 15
. Optimal error correction . . The posterior distribution of additional chains E ∗ conditioned on ∂ E is e K p τ E ∗ P ( E ∗ | ∂ E ) ∝ ∏ ij ij where E ∗ + E + C = C ∗ and C ∗ is trivial cycle while C is nontrivial one. The trivial cycle reads τ E ∗ ij τ E ij τ C ij = σ i σ j . . . . . Mapping to Spin-glass theory . . Summation over C ∗ yields probability of C conditioned on ∂ E . e K p τ C ij τ E ij σ i σ j = Z C ( K p ) ∑ P ( E ∗ | ∂ E ) ∝ ∑ ∏ P ( C | ∂ E ) = E ∗ + E + C = C ∗ { σ i } ij where Z L ( K p ) is the partition function of the Edwards-Anderson model. . . . . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 6 / 15
. Optimal error correction . . The posterior distribution of additional chains E ∗ conditioned on ∂ E is e K p τ E ∗ P ( E ∗ | ∂ E ) ∝ ∏ ij ij where E ∗ + E + C = C ∗ and C ∗ is trivial cycle while C is nontrivial one. The trivial cycle reads τ E ∗ ij τ E ij τ C ij = σ i σ j . . . . . Mapping to Spin-glass theory . . Summation over C ∗ yields probability of C conditioned on ∂ E . e K p τ C ij τ E ij σ i σ j = Z C ( K p ) ∑ P ( E ∗ | ∂ E ) ∝ ∑ ∏ P ( C | ∂ E ) = E ∗ + E + C = C ∗ { σ i } ij where Z L ( K p ) is the partition function of the Edwards-Anderson model. . . . . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 6 / 15
. Optimal error correction . . The posterior distribution of additional chains E ∗ conditioned on ∂ E is e K p τ E ∗ P ( E ∗ | ∂ E ) ∝ ∏ ij ij where E ∗ + E + C = C ∗ and C ∗ is trivial cycle while C is nontrivial one. The trivial cycle reads τ E ∗ ij τ E ij τ C ij = σ i σ j . . . . . Mapping to Spin-glass theory . . Summation over C ∗ yields probability of C conditioned on ∂ E . e K p τ C ij τ E ij σ i σ j = Z C ( K p ) ∑ P ( E ∗ | ∂ E ) ∝ ∑ ∏ P ( C | ∂ E ) = E ∗ + E + C = C ∗ { σ i } ij where Z L ( K p ) is the partition function of the Edwards-Anderson model. . . . . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 6 / 15
X XY XY Y X Φ Φ C C Y . How to identify the error correctablity? . . Compute the (finite but large-size) partition function with/without nontrivial cycles (Dennis 2002) { 1 ( ∃ C ) P ( C | ∂ E ) = Z C ( K p ) correctable ∑ Z ( K p ) = Z ( K p ) = Z C ( K p ) 1 / 4 uncorrectable . . . C . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 7 / 15
. My hope . . Compute the precise error thresholds in analytical way! . . . . T Possible analytical way? . . Without disorder, the duality is available T c Z ( K ) = λ N B Z ( K ∗ ) ( p ,T ) N N where exp( − 2 K ∗ ) = tanh K . K = K ∗ leads to the critical point. The duality is applicable if p 0 p Self dual (Ising, Potts models) Red: Nishimori line (1 / T = K p ) Transition occurs odd times. . . . . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 8 / 15
. My hope . . Compute the precise error thresholds in analytical way! . . . . T Possible analytical way? . . Without disorder, the duality is available T c Z ( K ) = λ N B Z ( K ∗ ) ( p ,T ) N N where exp( − 2 K ∗ ) = tanh K . K = K ∗ leads to the critical point. The duality is applicable if p 0 p Self dual (Ising, Potts models) Red: Nishimori line (1 / T = K p ) Transition occurs odd times. . . . . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 8 / 15
. Duality for spin glass: Nishimori and Nemoto (2002) . . Duality transformation with replica method estimates the location of the critical points from [ λ n ] = 1 → [log λ ] = 0, but it fails self-duality. = 1 ( 1 + e − 2 / T ) ( 1 + e 2 / T ) (1 − p ) log + p log 2 log 2 . which leads to p N = 0 . 1100 ... (cf. 0.10919(7) by MCMC). . . . T T c . Renormalization (Ohzeki 2009) . . On the renormalized system, the duality ( p ,T ) N N analysis leads to more precise value by [log λ ( s ) c ] = 0 as p N = 0 . 1092 ... . . . . p ? p 0 . . . . . . Masayuki Ohzeki (Kyoto University) AQC2015 2015/07/01 9 / 15
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