Quantum phase estimation is one of the key algorithms in the field of quantum computing, but up until now, only approximate expressions have been derived for the probability of error. We revisit these derivations, and find that by ensuring symmetry in the error definitions, an exact formula can be found. This new approach may also have value in solving other related problems in quantum computing, where an expected error is calculated. Expressions for two special cases of the formula are also developed, in the limit as the number of qubits in the quantum computer approaches infinity and in the limit as the extra added qubits to improve reliability goes to infinity. It is found that this formula is useful in validating computer simulations of the phase estimation procedure and in avoiding the overestimation of the number of qubits required in order to achieve a given reliability. This formula thus brings improved precision in the design of quantum computers.
Quantum phase estimation is one of the key algorithms in the field of quantum computing, but up until now, only approximate expressions have been derived for the probability of error. We revisit these derivations, and find that by ensuring symmetry in the error definitions, an exact formula can be found. This new approach may also have value in solving other related problems in quantum computing, where an expected error is calculated. Expressions for two special cases of the formula are also developed, in the limit as the number of qubits in the quantum computer approaches infinity and in the limit as the extra added qubits to improve reliability goes to infinity. It is found that this formula is useful in validating computer simulations of the phase estimation procedure and in avoiding the overestimation of the number of qubits required in order to achieve a given reliability. This formula thus brings improved precision in the design of quantum computers.
The accumulation of noise in quantum computers is the dominant issue stymieing the push of quantum algorithms beyond their classical counterparts.We do not expect to be able to afford the overhead required for quantum error correction in the next decade, so in the meantime we must rely on low-cost, unscalable error mitigation techniques to bring quantum computing to its full potential.In this paper we present a new error mitigation technique based on quantum phase estimation that can also reduce errors in expectation value estimation (e.g., for variational algorithms).The general idea is to apply phase estimation while effectively postselecting for the system register to be in the starting state, which allows us to catch and discard errors that knock us away from there.We refer to this technique as "verified phase estimation" (VPE) and show that it can be adapted to function without the use of control qubits in order to simplify the control circuitry for near-term implementations.Using VPE, we demonstrate the estimation of expectation values on numerical simulations of intermediate-scale quantum circuits with multiple orders of magnitude improvement over unmitigated estimation at near-term error rates (even after accounting for the additional complexity of phase estimation).Our numerical results suggest that VPE can mitigate against any single errors that might occur; i.e., the error in the estimated expectation values often scale as O(p 2 ), where p is the probability of an error occurring at any point in the circuit.This property reveals VPE as a practical technique for mitigating errors in near-term quantum experiments.
Abstract One of the main important features of the noisy intermediate-scale quantum(NISQ) era is the correct evaluation and consideration of errors. In this paper, we analyze the main sources of errors in current (IBM) quantum computers and we present a useful tool (TED-qc) designed to facilitate the total error probability expected for any quantum circuit. We propose this total error probability as the best way to estimate a lower bound for the fidelity in the NISQ era, avoiding the necessity of comparing the quantum calculations with any classical one. In order to contrast the robustness of our tool we compute the total error probability that may occur in three different quantum models: 1) the Ising model, 2) the Quantum-Phase Estimation (QPE), and 3) the Grover’s algorithm. For each model, the main quantities of interest are computed and benchmarked against the reference simulator’s results as a function of the error probability for a representative and statistically significant sample size. The analysis is satisfactory in more than the 99% of the cases. In addition, we study how error mitigation techniques are able to eliminate the noise induced during the measurement. These results have been calculated for the IBM quantum computers, but both the tool and the analysis can be easily extended to any other quantum computer.
Abstract This paper is concerned with the phase estimation algorithm in quantum computing, especially the scenarios where (1) the input vector is not an eigenvector; (2) the unitary operator is approximated by Trotter or Taylor expansion methods; (3) random approximations are used for the unitary operator. We characterize the probability of computing the phase values in terms of the consistency error, including the residual error, Trotter splitting error, or statistical mean-square error. In the first two cases, we show that in order to obtain the phase value with error less or equal to 2 − n and probability at least 1 − ϵ , the required number of qubits is t ⩾ n + log 2 + δ 2 2 ϵ Δ E 2 . The parameter δ quantifies the error associated with the inexact eigenvector and/or the unitary operator, and Δ E characterizes the spectral gap, i.e., the separation from the rest of the phase values. This analysis generalizes the standard result (Cleve et al 1998 Phys. Rev X 11 011020; Nielsen and Chuang 2002 Quantum Computation and Quantum Information ) by including these effects. More importantly, it shows that when δ < Δ E , the complexity remains the same. For the third case, we found a similar estimate, but the number of random steps has to be sufficiently large.
After the advent of Quantum Physics, new opportunities were created for research and development in the world of science and technology.Quantum computation is one such emerging technology which allows for notable speed-ups compared to conventional computational methods.Here a quantum bit simulator is presented.This simulator is an engineering work, and no deep understanding of quantum mechanics is required from the user.The well-known phase estimation algorithm is presented using this simulator.The qubit simulator will be a useful tool for the study of quantum computer circuits, quantum computing, and the development of new quantum algorithms
Trotter product formulas are a leading approach for Hamiltonian simulation on quantum computers, yet their practical performance has remained difficult to assess due to the challenge of accurately estimating the Trotter error. In this work, we develop new theoretical results, algorithms, and software tools that advance the state-of-the-art in Trotter error estimation by orders of magnitude in both scale and accuracy. On the theoretical side, we prove that in the asymptotic limit the error of a product formula depends on the diagonal elements of the Baker-Campbell-Hausdorff (BCH) error operator in the eigenbasis of the Hamiltonian, rather than its full spectral norm -- yielding an improved scaling for Hamiltonian simulation using product formulas. On the algorithmic side, we introduce a compact representation of the BCH expansion that reduces the number of commutators from O(n3) to O(n) for second-order, and from O(n5) to O(n2) for fourth-order formulas on n fragments, complemented by an importance sampling scheme to further reduce the computational cost. We provide implementations of these techniques in software and demonstrate their power on two applications: (i) X-ray absorption spectroscopy of an electronic Hamiltonian (Li4Mn2O) at up to 56 qubits using tensor networks; and (ii) vibronic dynamics of naphthalene at over 100 qubits using ML-MCTDH, where we find that naive analytical bounds overestimate the required number of Trotter steps by nearly five orders of magnitude. Our framework enables, for the first time, the accurate estimation of Trotter error at practically relevant system sizes, providing a foundation for fair algorithmic comparisons and rational design of product formulas.
We establish tight connections between entanglement entropy and the approximation error in Trotter-Suzuki product formulas for Hamiltonian simulation. Product formulas remain the workhorse of quantum simulation on near-term devices, yet standard error analyses yield worst-case bounds that can vastly overestimate the resources required for structured problems. For systems governed by geometrically local Hamiltonians with maximum entanglement entropy Smax across all bipartitions, we prove that the first-order Trotter error scales as O(t2Smaxpolylog(n)/r) rather than the worst-case O(t2n/r), where n is the system size and r is the number of Trotter steps. This yields improvements of Ω~(n2) for one-dimensional area-law systems and Ω~(n3/2) for two-dimensional systems. We extend these bounds to higher-order Suzuki formulas, where the improvement factor involves 2pS∗/2 for the p-th order formula. We further establish a separation result demonstrating that volume-law entangled systems fundamentally require Ω~(n) more Trotter steps than area-law systems to achieve the same precision. This separation is tight up to logarithmic factors. Our analysis combines Lieb-Robinson bounds for locality, tensor network representations for entanglement structure, and novel commutator-entropy inequalities that bound the expectation value of nested commutators by the Schmidt rank of the state. These results have immediate applications to quantum chemistry, condensed matter simulation, and resource estimation for fault-tolerant quantum computing.