量子错误缓解.pdf
Quantum error mitigation Ying Li, 2023 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Universal / digital / circuit-based quantum computer 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Classical computing with logic gates • Logic gates are operations on bits. • {NOT, AND} and {NOT, OR} are universal gate sets. https://en.wikipedia.org/wiki/Logic_gate 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Qubits and quantum gates Superposition state of a qubit: θ θ iϕ | ψ⟩ = cos | 0⟩ + e sin | 1⟩ 2 2 = θ cos 2 e iϕ θ sin 2 Universal gate sets: • Clifford gates + one non-Clifford {H, T, CNOT} or {H, T, CZ} (Realistic gate sets) • Toffoli and Hadamard {CCNOT, H} https://en.wikipedia.org/wiki/Quantum_logic_gate 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Universal quantum computer (Quantum Turing machine) Quantum circuit: Time |0i 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|0i T Initialisation Quantum gate 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 0 Quantum Computing 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|0i |0i 0 Classical Computing 0 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 Readout David Deutsch, Proceedings of the Royal Society A 400, 97-117 (1985) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Hardware Google's 54-qubit Sycamore chip https://www.nature.com/articles/d41586-019-03213-z • Superconducting qubits Solid, fast and scalable • Ion trap Accurate, light-matter interface • Photonics • Quantum dot • Neutral atoms • Majorana fermions • …… IBM Quantum System One IonQ's Aria system https://qiskit.org/documentation/qc_intro.html https://ionq.com/news/march-21-2022-ionqaria-coming-to-microsoft-azure-quantum 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP 1. Background 2. What is quantum error mitigation 3. Error-model-based approaches 4. Constraint-based approaches 5. Learning-based approaches 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP 1. Background 2. What is quantum error mitigation 3. Error-model-based approaches 4. Constraint-based approaches 5. Learning-based approaches 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Decoherence and memory time Microseconds Seconds Years Trapped ions Superconducting qubits 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Thousands of Years 30000 Years Resilience of classical information 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Classical error correction: 0=000010000… 1=111101111… Two types of errors on qubits: X Z Surface code: Protection against X errors Quantum error correction Protection against Z errors A. Yu. Kitaev, Annals Phys. 303, 2-30 (2003) Robert Raussendorf and Jim Harrington, Phys. Rev. Lett. 98, 190504 (2007) Austin G. Fowler, Ashley M. Stephens, and Peter Groszkowski, Phys. Rev. A 80, 052312 (2009) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Logical-qubit error rate (%) 4 3 L=3 L=5 L=7 L=9 L = 11 2 Threshold Surface-code threshold 1 0 0 Thresholds of surface code: 0.4 0.8 1.2 Two-qubit gate error rate (%) 0.75%, Robert Raussendorf and Jim Harrington, 2006 > 1%, David Wang, Austin Fowler and Lloyd Hollenberg, 2011 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP 1.6 Quantum fault tolerance Threshold theorem: • If the physical error rate is lower than the threshold, • the logical error rate can be suppressed to an arbitrarily low level, • and the number of physical qubits for encoding is polynomial in one over the logical error rate. Dorit Aharonov and Michael Ben-Or, arXiv:quant-ph/9611025 Emanuel Knill, Raymond Laflamme, and Wojciech H. Zurek, Science 279, 342-345 (1998) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Error rate over years https://nqit.ox.ac.uk/content/ion-traps 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Threshold error rate above 1% Gap #1 has been closed!!! Technology today: Error rate below 0.1% 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP log10(no. stabilising cycles) Fault-tolerant quantum computing and the qubit overhead 15 12 9 6 3 0 −5 log −4 10 (ga te −3 err or r ate −2 ) 0 1 2 le up) a c s ( log10 3 4 Encoding cost of surface code 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Fault-tolerant quantum computing Magic-state encoding YL, New J. Phys. 17, 023037 (2015) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Qubit cost of fault-tolerant quantum computing Joe O'Gorman and Earl T. Campbell, Phys. Rev. A 95, 032338 (2017) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Qubit number 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 FTQC 1,000 logical qubits 10-12 error rate QFT Gap #2 D-Wave 量⼦错误缓解 Google IBM 10 UCSB 0 10 -1 10 -2 10 Oxford, ion -3 Error rate 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP 10 -4 10 -5 Optical and network quantum computing Size 0[1] 1[2] 2[3,4] 3[4] 4[4] 4[5] 5[6] 31×31[6] 35 (1D)[7] Local error rate --0.1% 0.1% 0.2% 0.775% 0.825% 0.1% 0.1% Network error rate [1] YL, Peter C. Humphreys, Gabriel J. Mendoza, and Simon C. Benjamin, Phys. Rev. X 5, 041007 (2015) [2] YL, Sean D. Barrett, Thomas M. Stace, and Simon C. Benjamin, Phys. Rev. Lett. 105, 250502 (2010) [3] YL, Daniel Cavalcanti, and Leong Chuan Kwek, Phys. Rev. A 85, 062330 (2012) [4] YL and Simon C. Benjamin, New J. Phys. 14, 093008 (2012) [5] Naomi H. Nickerson, YL, and Simon C. Benjamin, Nat. Commun. 4, 1756 (2013) [6] YL and Simon C. Benjamin, Phys. Rev. A 94, 042303 (2016) [7] YL and Simon C. Benjamin, npj Quantum Inf. 4, 25 (2018) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP 0.001% 0.01% 1% 10% 30% 10% 10% 10% 0.1% Quantum computing on segmented chain YL and Simon C. Benjamin, npj Quantum Inf. 4, 25 (2018) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Linear optical quantum computing Intermediates states GHZ states (core qubits physically measured here) Building-block states 3D cluster state Stage-i Stage-ii Stage-iii BM GHZ BM N output modes M input modes Switch GHZ Loss ~ 4X10-4 Error ~ 3X10-6 GHZ Detectors ~ 106 other qubits 75% success A B BM A B BM qubit-3 Switch qubit-2 Switch qubit-1 BM GHZ 50% success YL, Peter C. Humphreys, Gabriel J. Mendoza, and Simon C. Benjamin, Phys. Rev. X 5, 041007 (2015) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Majorana fermion quantum computing YL, Phys. Rev. Lett. 117, 120403 (2016) YL, Phys. Rev. A 98, 012336 (2018) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP 1. Background 2. What is quantum error mitigation 3. Error-model-based approaches 4. Constraint-based approaches 5. Learning-based approaches 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP What is quantum error mitigation Quantum error mitigation refers to methods for • attaining the correct computing result using data from quantum circuits already affected by errors, • instead of preventing errors from happening. 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP What is quantum error mitigation • Hardware and control optimisation: Minimising physical errors • Quantum error correction: Minimising logical errors • Quantum error mitigation: Minimising the impact of errors 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP 1. Background 2. What is quantum error mitigation 3. Error-model-based approaches 4. Constraint-based approaches 5. Learning-based approaches 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Error extrapolation (Zero-noise extrapolation) Computation result Deliberately make the errors worse! ⟨X⟩3 ⟨X⟩1 ⟨X⟩ Quantum Error mitigation formula: ⟨X⟩2 em Increasing the error rate ef Error rate YL and Simon C. Benjamin, Phys. Rev. X 7, 021050 (2017) Kristan Temme, Sergey Bravyi, and Jay M. Gambetta, Phys. Rev. Lett. 119, 180509 (2017) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP ⟨X⟩ = F(⟨X⟩1, ⟨X⟩2, ⟨X⟩3, …) Error extrapolation (Zero-noise extrapolation) A. Kandala, K. Temme, A. D. Corcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta, Nature 567, 491 (2019) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Error extrapolation (Zero-noise extrapolation) • 26 spin 2D Ising spin lattice • Digital quantum simulation with Trotterisation • Circuit depth of 60 and 1080 CNOT gates Youngseok Kim et al., arXiv:2108.09197 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP General formalism of quantum error mitigation Error-free original circuit C: Original circuit C Variant circuits · · · C3 C2 C Noisy device Noisy device: U Ideal device: U 1 + QEM formula F (yC1 , yC2 , yC3 , ...) ef ⊗n = | 0⟩⟨0 | ef • TPCP map of the circuit: ℳ ef • Measured observable: O • Initial state: ρ • Error-free computation result: fC = Tr [O ef ℳ (ρ )] ef ef Noisy original circuit C: • Initial state: ρ Probability • TPCP map of the circuit: ℳ • Measured observable: O yC ! fC fC • Error-free computation result: yC = Tr [Oℳ (ρ)] Dayue Qin, Yanzhu Chen, and YL, arXiv:2112.06255 Dayue Qin, Xiaosi Xu, and YL, An overview of quantum error mitigation formulas, Chinese Phys. B 31 090306 (2022) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP General formalism of quantum error mitigation Original circuit C Variant circuits · · · C3 C2 C 1 Noisy variant circuit Ck: • Initial state: ρ • TPCP map of the circuit: ℳk • Measured observable: Ok Noisy device Noisy device: U Ideal device: U + QEM formula • Error-free computation result: yCk = Tr [Okℳk (ρ)] F (yC1 , yC2 , yC3 , ...) Probability Error mitigation formula: f′C = F(yC1, yC2, yC3, …) yC ! fC fC  Dayue Qin, Yanzhu Chen, and YL, arXiv:2112.06255 Dayue Qin, Xiaosi Xu, and YL, An overview of quantum error mitigation formulas, Chinese Phys. B 31 090306 (2022) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP General formalism of quantum error mitigation Original circuit C Variant circuits · · · C3 C2 C 1 • Error before error mitigation: yC − fC Noisy device Noisy device: U F (yC1 , yC2 , yC3 , ...) Probability Ideal device: U + QEM formula • Error after error mitigation: f′C − fC • Variances yC ! fC fC  Dayue Qin, Yanzhu Chen, and YL, arXiv:2112.06255 Dayue Qin, Xiaosi Xu, and YL, An overview of quantum error mitigation formulas, Chinese Phys. B 31 090306 (2022) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Quasi-probability decomposition and probabilistic error cancellation Quasi-probability decomposition: O ef ℳ (ρ ) = qkOkℳk (ρ) ∑ ef ef k Error mitigation formula: f′C = ∑ k qk yCk Probabilistic error cancellation (Monte Carlo): f′C = Cost × pk = | qk | /Cost, Cost = ∑ k | qk | , Variance ∝ Cost ∑ k sgn (qk) pk yCk 2   Kristan Temme, Sergey Bravyi, and Jay M. Gambetta, Phys. Rev. Lett. 119, 180509 (2017) Suguru Endo, Simon C. Benjamin, and YL, Phys. Rev. X 7 8, 031027 (2018) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Probabilistic error cancellation f′C = q1yC1 + q2yC2 + q3yC3 + ⋯ Classical data processing The original gate is replaced by a Clifford gate/Initialisation/ Measurement |0! Measurement outcome μ |0! Probability (a) Computing without error mitigation |0! Ideal value µ i1 ρk 2 i2 ρk 3 i3 ··· ··· ··· ··· ··· ··· ··· Qj1 ··· Qj2 ··· Qj3 Random numbers l = (i1 , j1 , k1 , . . . ) Measurement outcome μ Probability ρk 1 (i4 , i5 ) (b) Computing with error mitigation Cµeff (l, µ)  Kristan Temme, Sergey Bravyi, and Jay M. Gambetta, Phys. Rev. Lett. 119, 180509 (2017) Provable effectiveness: Suguru Endo, Simon C. Benjamin, and YL, Phys. Rev. X 7 8, 031027 (2018) The bias can be completely eliminated. 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Probabilistic error cancellation - Universal operation set Suguru Endo, Simon C. Benjamin, and YL, Phys. Rev. X 7 8, 031027 (2018) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Probabilistic error cancellation - Consistency Standard Quantum Theory Representation Gate Set Tomography Representation (Unknown) (Known) Initial state |ρ̄k !! Operation !!Q̄j |Ōi |ρ̄k "" = !!Q̂j |Ôi |ρ̂k "" !!Q̄j | Quantum circuit Initial state Operation |ρ̄(0) !! = S −1 |ρ̂(0) !! Ō (0) =S −1 |ρ̂k !! = S|ρ̄k !! Operation Ôi = S Ōi S −1 Gate Set Tomography Ōi Measured quantity Initial state Ô (0) S Measured quantity !!Q̄(0) | = !!Q̂(0) |S Quantum computing without error Measured quantity !!Q̂j | = !!Q̄j |S −1 Quasi-probability decomposition Initial state Operation !!Q̄(0) |Ō(0) |ρ̄(0) "" = !!Q̂ (0) |Ô (0) |ρ̂ (0) "" |ρ̂(0) !! = Ô(0) = Measured quantity !!Q̂ (0) |= ! qk |ρ̂k !! i ! qj !!Q̂j | k ! j qi Ôi Suguru Endo, Simon C. Benjamin, and YL, Phys. Rev. X 7 8, 031027 (2018) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP X π2 |0! |0 + 1! Mσi Probabilistic error cancellation σb Zφ (e) Two-qubit random circuits (f) One-qubit computation results Instance 1 MI GST: Gate Set Tomography Zπ Zφ Yπ GST circuits µ1 Instance 2 X π2 Decomposition formulas Q2 |0 + 1! Y π2 Classical computer Estimation of operations Q1 |0 + 1! µ2 MXGST data Random circuits Instance 3 Weighted average (Computation result) Q1 |0 + 1! M I +X 2 Q2 |0 + 1! Quantum computer µ3 R|0+1! µ!3 MY Computation data Z π2 µ4 Instance 4 (c) Controlled- φ -phase gate Q1 |0 + 1! M I +Y Q2 |0 + 1! M I +Z 2 R|0+i 1! MZ !X " = 21 (w1 µ1 + w2 µ2 + w3 µ3 µ3! + w4 µ4 µ4! µ4!! · · · ) Q2 |0 + 1! µ!4 0 Q1 = Cφ Zφ Y π2 Xπ 2 X− π2 Y− π2 Uφ Y π2 Xπ X− π2 Y− π2 µ!!4 2 2 R|ψ! QA1 or QA2 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP QEM 0.2 0.1 QA1 -0.05 0 !Z " 0.05 0.1 0.5 Bus resonator Q1 or Q2 Z φ−π 0.003±0.022 (g) Two-qubit computation results Q2 QA2 1 (d) Reset gate Z φ−π No QEM 0 -0.1 Chao Song, Jing Cui, H. Wang, J. Hao, H. Feng, and YL, Sci. Adv. 5, eaaw5686 (2019) R|0! 0.027±0.016 Prob. Zπ 0.2 (b) Qubit layout 0.1 Prob. Xπ Computation result Q1 |0 + 1! (a) Universal quantum error mitigation σd Zφ !X " = |0! Ideal Without QEM With G QEM 0 -0.5 0 π/4 π/2 φ 3π/4 π Probabilistic error cancellation Shuaining Zhang, Yao Lu, Kuan Zhang, Wentao Chen, YL, Jing-Ning Zhang, and Kihwan Kim, Nature Communications 11, 587 (2020) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Probabilistic error cancellation Ten superconducting qubits: Ewout van den Berg, Zlatko K. Minev, Abhinav Kandala, Kristan Temme, arXiv:2201.09866 Four ion-trap qubits: Wentao Chen, Shuaining Zhang, Jialiang Zhang, Xiaolu Su, Yao Lu, Kuan Zhang, Mu Qiao, Ying Li, Jing-Ning Zhang, Kihwan Kim, arXiv:2302.10436 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Cost of quantum error mitigation f′C = q1yC1 + q2yC2 + q3yC3 + ⋯ Monte Carlo summation: 2 Var ∝ ( | q1 | + | q2 | + | q3 | + ⋯) ≃ exp(4Np)  Np = Gate number × Error rate ≲ 1 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP 1. Background 2. What is quantum error mitigation 3. Error-model-based approaches 4. Constraint-based approaches 5. Learning-based approaches 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Constraint-based approaches • Quantum error correction — Stabiliser group • Symmetry verification • Purification of fermion correlations • Purification of quantum states • …… 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Symmetry-based quantum error mitigation The final state | ψ⟩ has the symmetry PS (projection operator), i.e. PS | ψ⟩ = | ψ⟩. State with noise ρn Error mitigated state ρem = PS ρnPS Tr (PS ρnPS) Sam McArdle, Xiao Yuan, and Simon Benjamin, Phys. Rev. Lett. 122, 180501 (2019) X. Bonet-Monroig, R. Sagastizabal, M. Singh, and T. E. O'Brien, Phys. Rev. A 98, 062339 (2018) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Purification and virtual distillation Purified state: ρ→ ρ 2 Tr ( ) ρ2 |+⟩ ρ ρ X O • Two copies of the state, qubit overhead • Controlled-swap operation must be error-free Bálint Koczor, Phys. Rev. X 11, 031057 (2021) William J. Huggins et al., Phys. Rev. X 11, 041036 (2021) Piotr Czarnik, Andrew Arrasmith, Lukasz Cincio, Patrick J. Coles, arXiv:2102.06056 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Purification and virtual distillation ρ = (1 − p) | ψ⟩⟨ψ | + p | ψ⊥⟩⟨ψ⊥ | 2 2 2 (1 − p) | ψ⟩⟨ψ | + p | ψ⊥⟩⟨ψ⊥ | ρ→ = 2 2 2 (1 − p) + p Tr (ρ ) ρ 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Dual-state purification (a) |0! A ρ B ρ̄ U (b) B † A U |0! † † (d) |b! (c) |0! R H B H H ⟨Za⟩0 ρρ̄ + ρ̄ρ ρρ̄ + ρ̄ρ † ⟨O⟩ = Tr O /Tr = B ( 2 ) ( 2 ) 1 + ⟨Xa⟩0 O = Z1 H Mingxia Huo and YL, Phys. Rev. A 105, 022427 (2022) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Dual-state purification !t = 0.5 t 10−1 X RX X 10−2 H H 10−3 H 4 6 8 10 12 No. of Qubits 4 RZ A 12 6 8 10 No. of Qubits (b) g=1 † DSP RX DSP&TP g=4 H DSP DSP&TP H g = 16 DSP H DSP&TP B (c) 0.0 Raw −0.4 DSP DSP&TP −0.8 −1.2 0.0 Energy (Hartree) (a) ! = 0.1 Experiment on ibmq_athens Energy (Hartree) Average-error rescaling factor Random circuit test 0.5 1.0 1.5 2.0 Distance (Angstrom) Mingxia Huo and YL, Phys. Rev. A 105, 022427 (2022) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP 0. −0. −0. −1. 1. Background 2. What is quantum error mitigation 3. Error-model-based approaches 4. Constraint-based approaches 5. Learning-based approaches 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Learning-based approach (a) A quantum circuit |0! H |0! H T ΛX H RX (φ) |0! H T Env. MZ • Loss(q) = difference between f′C and fC MZ • Minimise Loss(q) H MZ (b) The noise model ρi f′C = q1yC1 + q2yC2 + q3yC3 + ⋯ H RZ (θ) Sys. MZ S ΛZ |0! H T M1 Ri MN M2 Ef P2i−1 Ri P2i+1 Armands Strikis, Dayue Qin, Yanzhu Chen, Simon C. Benjamin, and YL, PRX Quantum 2, 040330 (2021)   中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Circuit frame (a) Task circuit |0! X R |0! R |0! R |0! R Gates on the frame are fixed. R R T H R S Gates in slots are variable. R H R T R |0! H Circuit frame (b) Complete slot setting |0! Ri |0! q1 q2 |0! Gj Ri+1 |0! |0! (c) Task-dependent slot setting Slots |0! |0! |0! |0! H H S |0! 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Quadratic error loss (a) A quantum circuit |0! H |0! H T H T MZ S ΛZ |0! ΛX H MZ RX (φ) H H (b) The noise model C∈ℝ MZ RZ (θ) |0! Lℝ = E [( f′C − fC) ] 2 T H MZ Ef M2 of circuits; MN all two-qubit gates are Clifford ℝ isMa1 subset Sys. ρ•i Env. • ℝ = ℂ, single-qubit gates are Clifford (Clifford sampling) • ℝ= , single-qubit gates are general unitaries (Unitary sampling) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP  𝕌 Armands Strikis, Dayue Qin, Yanzhu Chen, Simon C. Benjamin, and YL, PRX Quantum 2, 040330 (2021) errors are gate-independent) 2 • ( f′C − fC) is Hom(2,2) • L 1 = Lℂ 0.1 0 −0.2 0 Error (d) Unitary vs Clifford 1.5 Exp. data 1 LU × 100 G µn /µn 0 Unitary sampling and−0.2 Clifford 0sampling 0.2 Error (c) Moments 100 Hom(1,1) • fC and f′C areUnitary Clifford (Assumption: 10 Single-qubit-gate 4 8 12 nth-Order Demonstrated with four superconducting qubits; 0.2 0.5 0 0 Randomly generated circuit with up to 10 layers of two-qubit gates; L U = LC 0.5 1 LC × 100 1.5 Observable is a single-qubit Pauli operator.   𝕌 Zhen Wang, Yanzhu Chen, Zixuan Song, Dayue Qin, Hekang Li, Qiujiang Guo, H. Wang, Chao Song, and YL, Phys. Rev. Lett. 126, 080501 (2021) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Learning-based quantum error mitigation A parametrised error mitigation formula: f′C(λ) = F(yC1, yC2, …, λ) Find optimal λ by minimising the error loss Lℂ(λ) = E [(f′C − fC) ] 2 C∈ℝ fC can be evaluated on a classical computer Armands Strikis, Dayue Qin, Yanzhu Chen, Simon C. Benjamin, and YL, PRX Quantum 2, 040330 (2021)   中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Demonstration on IBMQ Circuit for experimental demonstration |0! |0! Error-3 H H Z Error-1 P H No. of circuits R Training Test 24 40 Error-2 (a) ibmq_5_yorktown 1.0 Error-free result Raw result 0.5 Error-mitigated result 0.0 (b) ibmq_ourense 1.0 (c) ibmq_santiago 1.0 0.5 0.5 0.0 0.0 −0.5 −0.5 −0.5 −1.0 0.0 −1.0 0.0 −1.0 0.0 0.5 1.0 θ/� 1.5 2.0 0.5 1.0 θ/� 1.5 2.0 0.5 1.0 θ/� 1.5 2.0 Armands Strikis, Dayue Qin, Yanzhu Chen, Simon C. Benjamin, and YL, PRX Quantum 2, 040330 (2021) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Application to Rabi frequency optimisation Initial 0.50 0.45 Qubit R 0.40 Rxy (θxy , φxy ) (c) Uphase Resonator LC × 100 (b) (a) Optimal Qi Rz (θz ) 0.35 Ωi Q1 0.30 Qj Ωj (d) Q2 Q3 Q4 −3 −2 −1 ∆Ω3 /2π (MHz) 0 −3 −2 −1 ∆Ω3 /2π (MHz) 0 Phase inversion Q1 |0! R Q2 |0! R R R R R Q3 |0! R R R R R Q4 |0! R R R R R R R R MZ R Infidelity 1-FU 0.070 0.065 0.600 0.055 0.050 Zhen Wang, Yanzhu Chen, Zixuan Song, Dayue Qin, Hekang Li, Qiujiang Guo, H. Wang, Chao Song, and YL, Phys. Rev. Lett. 126, 080501 (2021) 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP Phenomenological global depolarising error model Global depolarising error model: ℳ (ρ) = (1 − ϵ)[U](ρ) + ϵ2 −n ⊗n I Simplest error mitigation formula: f′C = ayC Piotr Czarnik, Andrew Arrasmith, Patrick J. Coles, Lukasz Cincio, Quantum 5, 592 (2021) The global depolarising error model is a better approximation when the gate number is larger. (a) Computation error without mitigation (b) Computation error after mitigation 2.0 100 Lraw /! 1.5 60 Count 100 Lmin /! 80 1.0 40 0 0 60 40 0.5 20  ! |Error|/! |Error|/! 80 ! N scaling to N scaling 20 200 400 600 800 1000 1200 Number of two-qubit gates NG 1400 0.0 0 200 400 600 800 1000 1200 Number of two-qubit gates NG 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP 1400 Dayue Qin, Yanzhu Chen, and Ying Li, arXiv:2112.06255 Qubit number 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 10 FTQC 1,000 logical qubits 10-12 error rate 0 10 -1 10 -2 10 -3 Error rate 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP 10 -4 10 -5 Qubit number 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 10 FTQC C E Q 50 qubits 2500 gates 0 10 -1 10 M E Q + QEM -2 10 -3 Error rate 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP 10 -4 10 -5 Thank you! 中国⼯程物理研究院研究⽣院 GRADUATE SCHOOL OF CAEP
量子错误缓解.pdf
