2000 character limit reached
The quantum adiabatic algorithm suppresses the proliferation of errors
Published 23 Apr 2024 in quant-ph, cond-mat.stat-mech, cond-mat.str-el, and physics.atom-ph | (2404.15397v1)
Abstract: The propagation of errors severely compromises the reliability of quantum computations. The quantum adiabatic algorithm is a physically motivated method to prepare ground states of classical and quantum Hamiltonians. Here, we analyze the proliferation of a single error event in the adiabatic algorithm. We give numerical evidence using tensor network methods that the intrinsic properties of adiabatic processes effectively constrain the amplification of errors during the evolution for geometrically local Hamiltonians. Our findings indicate that low energy states could remain attainable even in the presence of a single error event, which contrasts with results for error propagation in typical quantum circuits.
- D. Stilck França and R. García-Patrón, Limitations of optimization algorithms on noisy quantum devices, Nature Physics 17, 1221 (2021).
- G. González-García, R. Trivedi, and J. I. Cirac, Error Propagation in NISQ Devices for Solving Classical Optimization Problems, PRX Quantum 3, 040326 (2022).
- S. D. Mishra, M. Frías-Pérez, and R. Trivedi, Classically computing performance bounds on depolarized quantum circuits, PRX Quantum 5, 020317 (2024).
- R. Trivedi, A. F. Rubio, and J. I. Cirac, Quantum advantage and stability to errors in analogue quantum simulators, arXiv preprint arXiv:2212.04924 (2022).
- T. Kato, On the Adiabatic Theorem of Quantum Mechanics, Journal of the Physical Society of Japan 5, 435 (1950).
- A. Messiah, Quantum mechanics: Volume II (North-Holland Publishing Company Amsterdam, 1962).
- S. Jansen, M.-B. Ruskai, and R. Seiler, Bounds for the adiabatic approximation with applications to quantum computation, Journal of Mathematical Physics 48, 102111 (2007).
- M. H. S. Amin, Consistency of the adiabatic theorem, Physical Review Letters 102, 220401 (2009).
- T. Albash and D. A. Lidar, Adiabatic Quantum Computing, Reviews of Modern Physics 90, 015002 (2018), arXiv: 1611.04471.
- A. M. Childs, E. Farhi, and J. Preskill, Robustness of adiabatic quantum computation, Physical Review A 65, 012322 (2001).
- J. Roland and N. J. Cerf, Noise resistance of adiabatic quantum computation using random matrix theory, Physical Review A 71, 032330 (2005).
- J. Åberg, D. Kult, and E. Sjöqvist, Robustness of the adiabatic quantum search, Physical Review A 71, 060312 (2005).
- T. Albash and D. A. Lidar, Decoherence in adiabatic quantum computation, Physical Review A 91, 062320 (2015).
- For classical Hamiltonians encoding combinatorial optimization problems the energy error is typically equivalent to the approximation ratio.
- E. R. Anschuetz and B. T. Kiani, Quantum variational algorithms are swamped with traps, Nature Communications 13, 7760 (2022).
- G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Physical Review Letters 93, 040502 (2004).
- A depolarizing noise channel on the i𝑖iitalic_ith qubit 𝒩i(d)subscriptsuperscript𝒩𝑑𝑖\mathcal{N}^{(d)}_{i}caligraphic_N start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is equivalent to stochastic Pauli noise as ρ𝜌\displaystyle\rhoitalic_ρ →𝒩i(d)ρ[𝒩i(d)]†→absentsubscriptsuperscript𝒩𝑑𝑖𝜌superscriptdelimited-[]subscriptsuperscript𝒩𝑑𝑖†\displaystyle\rightarrow\mathcal{N}^{(d)}_{i}\rho[\mathcal{N}^{(d)}_{i}]^{\dagger}→ caligraphic_N start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ [ caligraphic_N start_POSTSUPERSCRIPT ( italic_d ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT (4) =(1−p)ρ+p3(σixρσix+σiyρσiy+σizρσiz).absent1𝑝𝜌𝑝3subscriptsuperscript𝜎𝑥𝑖𝜌subscriptsuperscript𝜎𝑥𝑖subscriptsuperscript𝜎𝑦𝑖𝜌subscriptsuperscript𝜎𝑦𝑖subscriptsuperscript𝜎𝑧𝑖𝜌subscriptsuperscript𝜎𝑧𝑖\displaystyle=(1-p)\rho+\frac{p}{3}(\sigma^{x}_{i}\rho\sigma^{x}_{i}+\sigma^{y% }_{i}\rho\sigma^{y}_{i}+\sigma^{z}_{i}\rho\sigma^{z}_{i}).= ( 1 - italic_p ) italic_ρ + divide start_ARG italic_p end_ARG start_ARG 3 end_ARG ( italic_σ start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ italic_σ start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ italic_σ start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_σ start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ italic_σ start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) . (5) where p/3𝑝3p/3italic_p / 3 is the probability to trigger one of the Pauli matrices.
- P. Jordan and E. P. Wigner, Über das Paulische Äquivalenzverbot (Springer, Berlin, Heidelberg, 1993) pp. 109–129.
- D. S. Franca and R. Garcia-Patron, Limitations of optimization algorithms on noisy quantum devices, arXiv:2009.05532 [quant-ph] (2020), arXiv: 2009.05532.
- U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of physics 326, 96 (2011).
- B. F. Schiffer, J. Tura, and J. I. Cirac, Adiabatic spectroscopy and a variational quantum adiabatic algorithm, PRX Quantum 3, 020347 (2022).
- A. Anshu, Concentration bounds for quantum states with finite correlation length on quantum spin lattice systems, New Journal of Physics 18, 083011 (2016).
- M. Fishman, S. White, and E. Stoudenmire, The ITensor software library for tensor network calculations, SciPost Physics Codebases 10.21468/scipostphyscodeb.4 (2022).
- K. Mølmer, Y. Castin, and J. Dalibard, Monte carlo wave-function method in quantum optics, Journal of the Optical Society of America B 10, 524 (1993).
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