Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2009.11151 · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Threshold theorem in isolated quantum dynamics with stochastic control errors

Manaka Okuyama, Kentaro Ohki, Masayuki Ohzeki

Published Dec 4, 2022• 2 citations
arXivHTMLPDF

Abstract

We investigate the effect of stochastic control errors in the time-dependent Hamiltonian on isolated quantum dynamics. The control errors are formulated as time-dependent stochastic noise in the Schrödinger equation. For a class of stochastic control errors, we establish a threshold theorem that provides a sufficient condition to obtain the target state, which should be determined in noiseless isolated quantum dynamics, as a relation between the number of measurements and noise strength. The theorem guarantees that if the sum of the noise strengths is less than the inverse of computational time, the target state can be obtained through a constant-order number of measurements. If the opposite is true, the number of measurements to guarantee obtaining the target state increases exponentially with computational time. Our threshold theorem can be applied to any isolated quantum dynamics such as quantum annealing and adiabatic quantum computation. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

DOI
10.1098/rsta.2021.0412

Related papers

Threshold theorem in isolated quantum dynamics with local stochastic control errors

Manaka Okuyama, Kentaro Ohki, Masayuki Ohzeki

We investigate the effect of local stochastic control errors in the time-dependent Hamiltonian on isolated quantum dynamics. The control errors are formulated as time-dependent stochastic noise in the Schrodinger equation. For any local stochastic control errors, we establish a threshold theorem that provides a sufficient condition to obtain the target state, which should be determined in noiseless isolated quantum dynamics, as a relation between the number of measurements required and noise strength. The theorem guarantees that if the sum of the noise strengths is less than the inverse of computational time, the target state can be obtained through a constant-order number of measurements. If the opposite is true, the required number of measurements increases exponentially with computational time. Our threshold theorem can be applied to any isolated quantum dynamics such as quantum annealing, adiabatic quantum computation, the quantum approximate optimization algorithm, and the quantum circuit model.

•Sep 23, 2020•1 citation

Threshold theorem in quantum annealing with deterministic analog control errors

Manaka Okuyama, Masayuki Ohzeki

We investigate the effect of deterministic analog control errors in the time-dependent Hamiltonian on isolated quantum dynamics. Deterministic analog control errors are formulated as time-dependent operators in the Schrodinger equation. We give an upper bound on the distance between two states in time evolution with and without deterministic analog control errors. As a result, we prove that, if the strength of deterministic analog control errors is less than the inverse of computational time, the final state in quantum dynamics without deterministic analog control errors can be obtained through a constant-order number of measurements in quantum dynamics with deterministic analog control errors.

arXiv:2301.13075•Jan 30, 2023•0 citations

Control limit for the quantum state preparation under stochastic control errors

Kohei Kobayashi

We investigate the effect of stochastic control errors on the Hamiltonian that controls a closed quantum system. Quantum information technologies require careful control for preparing a desired state used as an information resource. However, because the stochastic control errors inevitably appear in realistic situations, it is difficult to completely implement the control Hamiltonian. Under this error, the actual performance of quantum control is far away from the ideal one, and thus it is of great importance to evaluate the effect of the control errors. In this paper, we derive a lower bound of the fidelity between two closed quantum systems obeying the dynamics with and without errors. This bound reveals a reachable and unreachable set of the controlled quantum system under stochastic noises. Also, it is easily computable without considering the stochastic process and needing the full dynamics of the states. We demonstrate the actual performance of this bound via a simple control example. Furthermore, based on this result, we quantitatively evaluate the probability of obtaining the target state in the presence of control errors.

arXiv:2211.14862•Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences•Sep 1, 2023•0 citations

Level-Reduction and the Quantum Threshold Theorem

Panagiotis Aliferis

Computers have led society to the information age revolutionizing central aspects of our lives from production and communication to education and entertainment. There exist, however, important problems which are intractable with the computers available today and, experience teaches us, will remain so even with the more advanced computers we can envision for tomorrow. Quantum computers promise speedups to some of these important but classically intractable problems. Simulating physical systems, a problem of interest in a diverse range of areas from testing physical theories to understanding chemical reactions, and solving number factoring, a problem at the basis of cryptographic protocols that are used widely today on the internet, are examples of applications for which quantum computers, when built, will offer a great advantage over what is possible with classical computer technology. The construction of a quantum computer of sufficient scale to solve interesting problems is, however, especially challenging. The reason for this is that, by its very nature, operating a quantum computer will require the coherent control of the quantum state of a very large number of particles. Fortunately, the theory of quantum error correction and fault-tolerant quantum computation gives us confidence that such quantum states can be created, can be stored in memory and can also be manipulated provided the quantum computer can be isolated to a sufficient degree from sources of noise. One of the central results in the theory of fault-tolerant quantum computation, the quantum threshold theorem shows that a noisy quantum computer can accurately and efficiently simulate any ideal quantum computation provided that noise is weakly correlated and its strength is below a critical value known as the quantum accuracy threshold. This thesis provides a simpler and more transparent non-inductive proof of this theorem based on the concept of level reduction. This concept is also used in proving the quantum threshold theorem for coherent and leakage noise and for quantum computation by measurements. In addition, the proof provides a methodology which allows us to establish improved rigorous lower bounds on the value of the quantum accuracy threshold.

•Jan 1, 2007•7 citations

Mathematical foundation of quantum annealing

Satoshi Morita, Hidetoshi Nishimori

Quantum annealing is a generic name of quantum algorithms that use quantum-mechanical fluctuations to search for the solution of an optimization problem. It shares the basic idea with quantum adiabatic evolution studied actively in quantum computation. The present paper reviews the mathematical and theoretical foundations of quantum annealing. In particular, theorems are presented for convergence conditions of quantum annealing to the target optimal state after an infinite-time evolution following the Schrödinger or stochastic (Monte Carlo) dynamics. It is proved that the same asymptotic behavior of the control parameter guarantees convergence for both the Schrödinger dynamics and the stochastic dynamics in spite of the essential difference of these two types of dynamics. Also described are the prescriptions to reduce errors in the final approximate solution obtained after a long but finite dynamical evolution of quantum annealing. It is shown there that we can reduce errors significantly by an ingenious choice of annealing schedule (time dependence of the control parameter) without compromising computational complexity qualitatively. A review is given on the derivation of the convergence condition for classical simulated annealing from the view point of quantum adiabaticity using a classical-quantum mapping.

arXiv:0806.1859•Journal of Mathematical Physics•Dec 1, 2008•380 citations

Hamiltonian formulation of quantum error correction and correlated noise: Effects of syndrome extraction in the long-time limit

E. Novais, Eduardo R. Mucciolo, Harold U. Baranger

We analyze the long-time behavior of a quantum computer running a quantum error correction (QEC) code in the presence of a correlated environment. Starting from a Hamiltonian formulation of realistic noise models, and assuming that QEC is indeed possible, we find formal expressions for the probability of a given syndrome history and the associated residual decoherence encoded in the reduced density matrix. Systems with nonzero gate times (``long gates'') are included in our analysis by using an upper bound on the noise. In order to introduce the local error probability for a qubit, we assume that propagation of signals through the environment is slower than the QEC period (hypercube assumption). This allows an explicit calculation in the case of a generalized spin-boson model and a quantum frustration model. The key result is a dimensional criterion: If the correlations decay sufficiently fast, the system evolves toward a stochastic error model for which the threshold theorem of fault-tolerant quantum computation has been proven. On the other hand, if the correlations decay slowly, the traditional proof of this threshold theorem does not hold. This dimensional criterion bears many similarities to criteria that occur in the theory of quantum phase transitions.

arXiv:0710.1624•Physical Review A•Jul 9, 2008•24 citations
We investigate the effect of stochastic control errors in the time-dependent Hamiltonian on isolated quantum dynamics. The control errors are formulated as time-dependent stochastic noise in the Schrödinger equation. For a class of stochastic control errors, we establish a threshold theorem that provides a sufficient condition to obtain the target state, which should be determined in noiseless isolated quantum dynamics, as a relation between the number of measurements and noise strength. The theorem guarantees that if the sum of the noise strengths is less than the inverse of computational time, the target state can be obtained through a constant-order number of measurements. If the opposite is true, the number of measurements to guarantee obtaining the target state increases exponentially with computational time. Our threshold theorem can be applied to any isolated quantum dynamics such as quantum annealing and adiabatic quantum computation. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
We investigate the effect of local stochastic control errors in the time-dependent Hamiltonian on isolated quantum dynamics. The control errors are formulated as time-dependent stochastic noise in the Schrodinger equation. For any local stochastic control errors, we establish a threshold theorem that provides a sufficient condition to obtain the target state, which should be determined in noiseless isolated quantum dynamics, as a relation between the number of measurements required and noise strength. The theorem guarantees that if the sum of the noise strengths is less than the inverse of computational time, the target state can be obtained through a constant-order number of measurements. If the opposite is true, the required number of measurements increases exponentially with computational time. Our threshold theorem can be applied to any isolated quantum dynamics such as quantum annealing, adiabatic quantum computation, the quantum approximate optimization algorithm, and the quantum circuit model.
We investigate the effect of deterministic analog control errors in the time-dependent Hamiltonian on isolated quantum dynamics. Deterministic analog control errors are formulated as time-dependent operators in the Schrodinger equation. We give an upper bound on the distance between two states in time evolution with and without deterministic analog control errors. As a result, we prove that, if the strength of deterministic analog control errors is less than the inverse of computational time, the final state in quantum dynamics without deterministic analog control errors can be obtained through a constant-order number of measurements in quantum dynamics with deterministic analog control errors.
We investigate the effect of stochastic control errors on the Hamiltonian that controls a closed quantum system. Quantum information technologies require careful control for preparing a desired state used as an information resource. However, because the stochastic control errors inevitably appear in realistic situations, it is difficult to completely implement the control Hamiltonian. Under this error, the actual performance of quantum control is far away from the ideal one, and thus it is of great importance to evaluate the effect of the control errors. In this paper, we derive a lower bound of the fidelity between two closed quantum systems obeying the dynamics with and without errors. This bound reveals a reachable and unreachable set of the controlled quantum system under stochastic noises. Also, it is easily computable without considering the stochastic process and needing the full dynamics of the states. We demonstrate the actual performance of this bound via a simple control example. Furthermore, based on this result, we quantitatively evaluate the probability of obtaining the target state in the presence of control errors.
Computers have led society to the information age revolutionizing central aspects of our lives from production and communication to education and entertainment. There exist, however, important problems which are intractable with the computers available today and, experience teaches us, will remain so even with the more advanced computers we can envision for tomorrow. Quantum computers promise speedups to some of these important but classically intractable problems. Simulating physical systems, a problem of interest in a diverse range of areas from testing physical theories to understanding chemical reactions, and solving number factoring, a problem at the basis of cryptographic protocols that are used widely today on the internet, are examples of applications for which quantum computers, when built, will offer a great advantage over what is possible with classical computer technology. The construction of a quantum computer of sufficient scale to solve interesting problems is, however, especially challenging. The reason for this is that, by its very nature, operating a quantum computer will require the coherent control of the quantum state of a very large number of particles. Fortunately, the theory of quantum error correction and fault-tolerant quantum computation gives us confidence that such quantum states can be created, can be stored in memory and can also be manipulated provided the quantum computer can be isolated to a sufficient degree from sources of noise. One of the central results in the theory of fault-tolerant quantum computation, the quantum threshold theorem shows that a noisy quantum computer can accurately and efficiently simulate any ideal quantum computation provided that noise is weakly correlated and its strength is below a critical value known as the quantum accuracy threshold. This thesis provides a simpler and more transparent non-inductive proof of this theorem based on the concept of level reduction. This concept is also used in proving the quantum threshold theorem for coherent and leakage noise and for quantum computation by measurements. In addition, the proof provides a methodology which allows us to establish improved rigorous lower bounds on the value of the quantum accuracy threshold.
Quantum annealing is a generic name of quantum algorithms that use quantum-mechanical fluctuations to search for the solution of an optimization problem. It shares the basic idea with quantum adiabatic evolution studied actively in quantum computation. The present paper reviews the mathematical and theoretical foundations of quantum annealing. In particular, theorems are presented for convergence conditions of quantum annealing to the target optimal state after an infinite-time evolution following the Schrödinger or stochastic (Monte Carlo) dynamics. It is proved that the same asymptotic behavior of the control parameter guarantees convergence for both the Schrödinger dynamics and the stochastic dynamics in spite of the essential difference of these two types of dynamics. Also described are the prescriptions to reduce errors in the final approximate solution obtained after a long but finite dynamical evolution of quantum annealing. It is shown there that we can reduce errors significantly by an ingenious choice of annealing schedule (time dependence of the control parameter) without compromising computational complexity qualitatively. A review is given on the derivation of the convergence condition for classical simulated annealing from the view point of quantum adiabaticity using a classical-quantum mapping.
We analyze the long-time behavior of a quantum computer running a quantum error correction (QEC) code in the presence of a correlated environment. Starting from a Hamiltonian formulation of realistic noise models, and assuming that QEC is indeed possible, we find formal expressions for the probability of a given syndrome history and the associated residual decoherence encoded in the reduced density matrix. Systems with nonzero gate times (``long gates'') are included in our analysis by using an upper bound on the noise. In order to introduce the local error probability for a qubit, we assume that propagation of signals through the environment is slower than the QEC period (hypercube assumption). This allows an explicit calculation in the case of a generalized spin-boson model and a quantum frustration model. The key result is a dimensional criterion: If the correlations decay sufficiently fast, the system evolves toward a stochastic error model for which the threshold theorem of fault-tolerant quantum computation has been proven. On the other hand, if the correlations decay slowly, the traditional proof of this threshold theorem does not hold. This dimensional criterion bears many similarities to criteria that occur in the theory of quantum phase transitions.