Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
162 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Early Fault-Tolerant Quantum Computing (2311.14814v1)

Published 24 Nov 2023 in quant-ph

Abstract: Over the past decade, research in quantum computing has tended to fall into one of two camps: near-term intermediate scale quantum (NISQ) and fault-tolerant quantum computing (FTQC). Yet, a growing body of work has been investigating how to use quantum computers in transition between these two eras. This envisions operating with tens of thousands to millions of physical qubits, able to support fault-tolerant protocols, though operating close to the fault-tolerant threshold. Two challenges emerge from this picture: how to model the performance of devices that are continually improving and how to design algorithms to make the most use of these devices? In this work we develop a model for the performance of early fault-tolerant quantum computing (EFTQC) architectures and use this model to elucidate the regimes in which algorithms suited to such architectures are advantageous. As a concrete example, we show that, for the canonical task of phase estimation, in a regime of moderate scalability and using just over one million physical qubits, the ``reach'' of the quantum computer can be extended (compared to the standard approach) from 90-qubit instances to over 130-qubit instances using a simple early fault-tolerant quantum algorithm, which reduces the number of operations per circuit by a factor of 100 and increases the number of circuit repetitions by a factor of 10,000. This clarifies the role that such algorithms might play in the era of limited-scalability quantum computing.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (128)
  1. Richard P. Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 21:467–488, 1982.
  2. Rapid solution of problems by quantum computation. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 439:553–558, 12 1992.
  3. Lov K. Grover. Quantum mechanics helps in searching for a needle in a haystack. Physical Review Letters, 79:325, 7 1997.
  4. Quantum complexity theory. In Proceedings of the twenty-fifth annual ACM symposium on Theory of computing, pages 11–20, 1993.
  5. Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Computing, 26:1484–1509, 1997.
  6. Peter W. Shor. Scheme for reducing decoherence in quantum computer memory. Physical Review A, 52:R2493, 10 1995.
  7. A. M. Steane. Active stabilization, quantum computation, and quantum state synthesis. Physical Review Letters, 78:2252, 3 1997.
  8. Perfect quantum error correcting code. Physical Review Letters, 77:198, 7 1996.
  9. Concatenated quantum codes. 8 1996.
  10. Quantum error correction and orthogonal geometry. Physical Review Letters, 78:405, 1 1997.
  11. A further requirement is that realistic assumptions be made on the noise model.
  12. P.W. Shor. Fault-tolerant quantum computation. Proceedings of 37th Conference on Foundations of Computer Science, pages 56–65.
  13. Fault-tolerant quantum computation with long-range correlated noise. Physical Review Letters, 96:050504, 2 2006.
  14. Fault-tolerant quantum computation with constant error rate. https://doi.org/10.1137/S0097539799359385, 38:1207–1282, 7 2008.
  15. A. Yu Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303:2–30, 1 2003.
  16. Simple proof of fault tolerance in the graph-state model. Physical Review A, 73(3):032308, 2006.
  17. Fault-tolerant quantum computation against biased noise. Physical Review A - Atomic, Molecular, and Optical Physics, 78:052331, 11 2008.
  18. Accuracy threshold for postselected quantum computation. Quantum Information & Computation, 8(3):181–244, 2008.
  19. Quantum accuracy threshold for concatenated distance-3 code. Quantum Information and Computation, 6:97–165, 3 2006.
  20. Surface codes: Towards practical large-scale quantum computation. Phys. Rev. A, 86:032324, Sep 2012.
  21. Fault tolerance of quantum low-density parity check codes with sublinear distance scaling. Physical Review A - Atomic, Molecular, and Optical Physics, 87:020304, 2 2013.
  22. Quantum low-density parity-check codes. PRX Quantum, 2:040101, 12 2021.
  23. Low-overhead fault-tolerant quantum computing using long-range connectivity. Science Advances, 8:1717, 5 2022.
  24. Neutral atom quantum computing hardware: performance and end-user perspective. EPJ Quantum Technology 2023 10:1, 10:1–26, 8 2023.
  25. Xiao Feng Shi. Quantum logic and entanglement by neutral rydberg atoms: methods and fidelity. Quantum Science and Technology, 7:023002, 4 2022.
  26. Demonstration of a small programmable quantum computer with atomic qubits. Nature 2016 536:7614, 536:63–66, 8 2016.
  27. Realization of a programmable two-qubit quantum processor. Nature Physics 2009 6:1, 6:13–16, 11 2009.
  28. Demonstration of a fundamental quantum logic gate. Physical Review Letters, 75:4714, 12 1995.
  29. Optimal control of superconducting gmon qubits using pontryagin’s minimum principle: Preparing a maximally entangled state with singular bang-bang protocols. Physical Review A, 97:062343, 6 2018.
  30. Qubit architecture with high coherence and fast tunable coupling. Physical Review Letters, 113:220502, 11 2014.
  31. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5:4213, 9 2014.
  32. Differentiable learning of quantum circuit born machines. Physical Review A, 98:062324, 12 2018.
  33. Quantum generative adversarial networks. Physical Review A, 98:012324, 7 2018.
  34. A quantum approximate optimization algorithm. arXiv preprint arXiv:1411.4028, 2014.
  35. Error mitigation for short-depth quantum circuits. Physical Review Letters, 119:180509, 11 2017.
  36. Error mitigation with clifford quantum-circuit data. Quantum, 5:592, 2021.
  37. Digital zero noise extrapolation for quantum error mitigation. pages 306–316. IEEE, 10 2020.
  38. Virtual distillation for quantum error mitigation. Physical Review X, 11:041036, 11 2021.
  39. Probabilistic error cancellation with sparse pauli–lindblad models on noisy quantum processors. Nature Physics, 19:1116–1121, 8 2023.
  40. John Preskill. Quantum computing in the nisq era and beyond. Quantum, 2:79, 2018.
  41. Evidence for the utility of quantum computing before fault tolerance. Nature 2023 618:7965, 618:500–505, 6 2023.
  42. Effective quantum volume, fidelity and computational cost of noisy quantum processing experiments. 6 2023.
  43. Efficient tensor network simulation of ibm’s eagle kicked ising experiment. 6 2023.
  44. Fast classical simulation of evidence for the utility of quantum computing before fault tolerance. arXiv preprint arXiv:2306.16372, 2023.
  45. Measurements as a roadblock to near-term practical quantum advantage in chemistry: resource analysis. Physical Review Research, 4(3):033154, 2022.
  46. Quantum supremacy through the quantum approximate optimization algorithm. arXiv preprint arXiv:1602.07674, 2016.
  47. Evidence of scaling advantage for the quantum approximate optimization algorithm on a classically intractable problem. arXiv preprint arXiv:2308.02342, 2023.
  48. Sampling frequency thresholds for the quantum advantage of the quantum approximate optimization algorithm. npj Quantum Information 2023 9:1, 9:1–10, 7 2023.
  49. Accelerated variational quantum eigensolver. Physical review letters, 122(14):140504, 2019.
  50. Quantum filter diagonalization: Quantum eigendecomposition without full quantum phase estimation. arXiv preprint arXiv:1909.08925, 2019.
  51. Minimizing estimation runtime on noisy quantum computers. PRX Quantum, 2(1):010346, 2021.
  52. Low depth algorithms for quantum amplitude estimation. Quantum, 6:745, 2022.
  53. Real-time evolution for ultracompact hamiltonian eigenstates on quantum hardware. PRX Quantum, 3(2):020323, 2022.
  54. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. Quantum, 7:1167, 2023.
  55. Ground-state preparation and energy estimation on early fault-tolerant quantum computers via quantum eigenvalue transformation of unitary matrices. PRX Quantum, 3(4):040305, 2022.
  56. State preparation boosters for early fault-tolerant quantum computation. Quantum, 6:829, oct 2022.
  57. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. arXiv preprint arXiv:2211.11973, 2022.
  58. Exact and efficient lanczos method on a quantum computer. Quantum, 7:1018, 2023.
  59. Quantum algorithms revisited. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 454(1969):339–354, 1998.
  60. Earl T Campbell. Early fault-tolerant simulations of the hubbard model. Quantum Science and Technology, 7(1):015007, 2021.
  61. Computing ground state properties with early fault-tolerant quantum computers. Quantum, 6:761, 9 2021.
  62. Lin Lin and Yu Tong. Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers. PRX Quantum, 3:010318, Feb 2022.
  63. Randomized quantum algorithm for statistical phase estimation. Physical Review Letters, 129(3):030503, 2022.
  64. Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation. PRX Quantum, 4:020331, 4 2023.
  65. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. 9 2022.
  66. Faster ground state energy estimation on early fault-tolerant quantum computers via rejection sampling. 4 2023.
  67. Fault-tolerant quantum computation with constant error rate. SIAM Journal on Computing, 38(4):1207–1282, 2008.
  68. Christof Zalka. Threshold estimate for fault tolerant quantum computation. arXiv preprint quant-ph/9612028, 1996.
  69. Topological quantum memory. Journal of Mathematical Physics, 43(9):4452–4505, 2002.
  70. Google Quantum AI. Suppressing quantum errors by scaling a surface code logical qubit. 2022.
  71. Surface code quantum computing with error rates over 1%. Physical Review A, 83(2):020302, 2011.
  72. Austin G Fowler. Proof of finite surface code threshold for matching. Physical review letters, 109(18):180502, 2012.
  73. Quantum logic with spin qubits crossing the surface code threshold. Nature, 601(7893):343–347, Jan 2022.
  74. Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography. Nature Communications, 8(1):14485, Feb 2017.
  75. Demonstration of fault-tolerant universal quantum gate operations. Nature, 605(7911):675–680, May 2022.
  76. Fault-tolerant operation of a quantum error-correction code. 2020.
  77. Daniel Gottesman. Stabilizer codes and quantum error correction. 1997.
  78. A fault-tolerant one-way quantum computer. Annals of Physics, 321(9):2242–2270, sep 2006.
  79. Tunable superconducting qubits with flux-independent coherence. Physical Review Applied, 8:044003, 10 2017.
  80. Entangling an arbitrary pair of qubits in a long ion crystal. Physical Review A, 98:032318, 9 2018.
  81. Pulse optimization for high-precision motional-mode characterization in trapped-ion quantum computers. 7 2023.
  82. Limitations in quantum computing from resource constraints. PRX Quantum, 2(4), nov 2021.
  83. Marco Fellous-Asiani. The resource cost of large scale quantum computing. 2021.
  84. Fault-tolerant resource estimate for quantum chemical simulations: Case study on li-ion battery electrolyte molecules. Physical Review Research, 4(2):023019, 2022.
  85. Reliably assessing the electronic structure of cytochrome p450 on today’s classical computers and tomorrow’s quantum computers. Proceedings of the National Academy of Sciences, 119(38):e2203533119, 2022.
  86. Assessing requirements to scale to practical quantum advantage. arXiv preprint arXiv:2211.07629, 2022.
  87. Daniel Litinski. Magic state distillation: Not as costly as you think. Quantum, 3:205, 2019.
  88. Transversal injection: A method for direct encoding of ancilla states for non-clifford gates using stabiliser codes. arXiv preprint arXiv:2211.10046, 2022.
  89. Partially fault-tolerant quantum computing architecture with error-corrected clifford gates and space-time efficient analog rotations. arXiv preprint arXiv:2303.13181, 2023.
  90. Robust calibration of a universal single-qubit gate set via robust phase estimation. Physical Review A, 92(6):062315, 2015.
  91. On proving the robustness of algorithms for early fault-tolerant quantum computers. arXiv preprint arXiv:2209.11322, 2022.
  92. On low-depth quantum algorithms for robust multiple-phase estimation. arXiv preprint arXiv:2303.08099, 2023.
  93. Especially in the case where the quantum computation is rate-limited by magic state distillation, the computational qubits would be required to idle without accruing error while waiting for T gates or Toffoli gates to be teleported into the computation.
  94. In the case where the tolerable circuit error rate approaches 1, pL≤(1/GC)⁢ln⁡(1/(1−pC))subscript𝑝𝐿1subscript𝐺𝐶11subscript𝑝𝐶p_{L}\leq(1/G_{C})\ln(1/(1-p_{C}))italic_p start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT ≤ ( 1 / italic_G start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ) roman_ln ( 1 / ( 1 - italic_p start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ) ) can be used as a tighter bound.
  95. Quantum amplitude amplification and estimation. Contemporary Mathematics, 305:53–74, 2002.
  96. Low overhead quantum computation using lattice surgery. arXiv preprint arXiv:1808.06709, 2018.
  97. Efficient quantum computation of molecular forces and other energy gradients. Physical Review Research, 4(4):043210, 2022.
  98. Modeling the performance of early fault-tolerant quantum algorithms. arXiv preprint arXiv:2306.17235, 2023.
  99. On low-depth algorithms for quantum phase estimation. arXiv preprint arXiv:2302.02454, 2023.
  100. Heisenberg-limited quantum phase estimation of multiple eigenvalues with few control qubits. Quantum, 6:830, 2022.
  101. Simultaneous estimation of multiple eigenvalues with short-depth quantum circuit on early fault-tolerant quantum computers. arXiv preprint arXiv:2303.05714, 2023.
  102. Estimating eigenenergies from quantum dynamics: A unified noise-resilient measurement-driven approach. arXiv preprint arXiv:2306.01858, 2023.
  103. Computing Ground State Properties with Early Fault-Tolerant Quantum Computers. Quantum, 6:761, July 2022.
  104. A multireference quantum krylov algorithm for strongly correlated electrons. Journal of chemical theory and computation, 16(4):2236–2245, 2020.
  105. Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision. arXiv, 2022.
  106. Low-depth gaussian state energy estimation. arXiv preprint arXiv:2309.16790, 2023.
  107. Single-ancilla ground state preparation via lindbladians. arXiv preprint arXiv:2308.15676, 2023.
  108. Foundations for bayesian inference with engineered likelihood functions for robust amplitude estimation. Journal of Mathematical Physics, 63(5), 2022.
  109. Robust ground-state energy estimation under depolarizing noise. arXiv preprint arXiv:2307.11257, 2023.
  110. Quantum algorithm for credit valuation adjustments. New Journal of Physics, 24(2):023036, 2022.
  111. Reducing runtime and error in vqe using deeper and noisier quantum circuits. arXiv preprint arXiv:2110.10664, 2021.
  112. Low-depth amplitude estimation on a trapped-ion quantum computer. Physical Review Research, 4(3):033034, 2022.
  113. Rolando D Somma. Quantum eigenvalue estimation via time series analysis. New Journal of Physics, 21(12):123025, 2019.
  114. Hamiltonian simulation using linear combinations of unitary operations. arXiv preprint arXiv:1202.5822, 2012.
  115. Earl Campbell. Random compiler for fast hamiltonian simulation. Physical review letters, 123(7):070503, 2019.
  116. Fault-tolerant quantum computation of molecular observables. arXiv preprint arXiv:2303.14118, 2023.
  117. Optimal quantum measurements of expectation values of observables. Physical Review A, 75(1):012328, 2007.
  118. Stochastic quantum krylov protocol with double-factorized hamiltonians. Physical Review A, 107(3):032414, 2023.
  119. Quantifying t-gate-count improvements for ground-state-energy estimation with near-optimal state preparation. Physical Review A, 107(4):L040601, 2023.
  120. When to reject a ground state preparation algorithm. arXiv preprint arXiv:2212.09492, 2022.
  121. Quantum computation and quantum information: 10th anniversary edition. 2011.
  122. Quantum error mitigation. arXiv preprint arXiv:2210.00921, 2022.
  123. Evidence for the utility of quantum computing before fault tolerance. Nature, 618:500–505, 06 2023.
  124. Robust measurement of wave function topology on NISQ quantum computers. Quantum, 7:987, apr 2023.
  125. Reducing the cost of energy estimation in the variational quantum eigensolver algorithm with robust amplitude estimation. arXiv preprint arXiv:2203.07275, 2022.
  126. Fault-tolerant resource estimate for quantum chemical simulations: Case study on li-ion battery electrolyte molecules. Physical Review Research, 4(2), apr 2022.
  127. Error mitigation for universal gates on encoded qubits. Phys. Rev. Lett., 127:200505, Nov 2021.
  128. Quantum error mitigation as a universal error reduction technique: Applications from the nisq to the fault-tolerant quantum computing eras. PRX Quantum, 3:010345, Mar 2022.
Citations (14)
View on Semantic Scholar

Summary

  • The paper proposes a scalability model that quantifies error scaling and qubit performance in early fault-tolerant systems.
  • The paper demonstrates enhanced phase estimation capabilities, expanding computational reach from 90-qubit to 130-qubit instances with significant reductions in operations.
  • The paper underscores the potential for novel quantum algorithms to exploit EFTQC advances and achieve practical applications in quantum simulation, chemistry, and optimization.

Insights on Early Fault-Tolerant Quantum Computing

The paper "Early Fault-Tolerant Quantum Computing" discusses the progression of quantum computing from the NISQ era to the anticipated fault-tolerant quantum computing era, with a focus on the intermediate stage known as Early Fault-Tolerant Quantum Computing (EFTQC). This paper aims to model EFTQC systems, which operate on a scale between thousands to millions of physical qubits, using near-fault-tolerant protocols. The authors emphasize the need for quantum algorithms that make these systems practical. Specifically, they develop a model that elucidates the advantages of quantum algorithms in this regime, highlighting a case paper on phase estimation.

Key Developments and Numerical Insights

  1. Scalability Model: The authors propose a scalability model to track the performance of quantum hardware as it transitions from NISQ to EFTQC and eventually to FTQC. This model is designed to reflect how error rates scale with the number of physical qubits and the threshold error rates required by quantum error correction codes.
  2. Phase Estimation: As a concrete example, the paper investigates phase estimation, a critical quantum computing task, demonstrating how early fault-tolerant algorithms can extend the capability of quantum systems. Remarkably, using just over one million physical qubits, the paper shows an increase in the "reach" of quantum computing from handling 90-qubit instances to over 130-qubit instances. This improvement comes with a reduction in the number of operations by 100-fold and an increase in repetition by 10,000-fold.
  3. Algorithm and Model Implications: The paper posits that new algorithms tailored for EFTQC can significantly improve quantum computing tasks beyond current capabilities, provided there are advancements in gate operations and error rates. The scalability model also suggests that optimal performance will depend on maintaining sub-threshold error rates as the system scales.

Theoretical and Practical Implications

The paper carries substantial implications for both theory and practice:

  • Theoretical Advancements: By providing a quantitative framework for assessing EFTQC systems, the research contributes to a better understanding of how error rates and algorithm designs impact quantum computing capabilities during the transition to fault-tolerance.
  • Practical Realizations: Practically, the reduction in operations per circuit and the enhanced scalability demonstrate the potential for running larger, more complex quantum computations with limited qubit resources. This has significant implications for near-term applications in quantum chemistry, optimization, and quantum simulations.
  • Future Prospects in Quantum Algorithms: The paper highlights the necessity for further algorithmic innovations that align with the scalable capacity of EFTQC hardware. These developments could pave the way toward utility-scale quantum advantage in various fields.

Speculations for Future Research

Given the current trajectory, future research might focus on refining error-correction techniques and scalability models to accommodate new hardware architectures. Further theoretical exploration in robust quantum algorithm development, tailored specifically for early fault-tolerant systems, could also accelerate quantum applications' industrial and scientific utility.

Overall, this paper underscores the transitional challenges and opportunities between the NISQ and FTQC eras, marking a pivotal step toward realizing practical quantum advantage. The insights gained here emphasize the importance of strategic algorithm-hardware alignment to unlock the full potential of early fault-tolerant quantum systems.

File Document Download Save Streamline Icon: https://streamlinehq.com PDF File Document Download Save Streamline Icon: https://streamlinehq.com Markdown
Youtube Logo Streamline Icon: https://streamlinehq.com

YouTube