The rapid development of noisy intermediate-scale quantum (NISQ) devices has raised the question of whether or not these devices will find commercial use. Unfortunately, a major shortcoming of many proposed NISQ-amenable algorithms, such as the variational quantum eigensolver (VQE), is coming into view: the algorithms require too many independent quantum measurements to solve practical problems in a reasonable amount of time. This motivates the central question of our work: how might we speed up such algorithms in spite of the impact of error on NISQ computations? We demonstrate on quantum hardware that the estimation of expectation values, a core subroutine of many quantum algorithms including VQE, can be improved in terms of precision and accuracy by using a technique we call Robust Amplitude Estimation. Consequently, this method reduces the runtime to achieve the same mean-squared error compared to the standard prepare-and-measure estimation method. The surprising result is that by using deeper, and therefore more error-prone, quantum circuits, we realize more accurate quantum computations in less time. As the quality of quantum devices improves, this method will provide a proportional reduction in estimation runtime. This technique may be used to speed up quantum computations into the regime of early fault-tolerant quantum computation and aid in the realization of quantum advantage.
The rapid development of noisy intermediate-scale quantum (NISQ) devices has raised the question of whether or not these devices will find commercial use. Unfortunately, a major shortcoming of many proposed NISQ-amenable algorithms, such as the variational quantum eigensolver (VQE), is coming into view: the algorithms require too many independent quantum measurements to solve practical problems in a reasonable amount of time. This motivates the central question of our work: how might we speed up such algorithms in spite of the impact of error on NISQ computations? We demonstrate on quantum hardware that the estimation of expectation values, a core subroutine of many quantum algorithms including VQE, can be improved in terms of precision and accuracy by using a technique we call Robust Amplitude Estimation. Consequently, this method reduces the runtime to achieve the same mean-squared error compared to the standard prepare-and-measure estimation method. The surprising result is that by using deeper, and therefore more error-prone, quantum circuits, we realize more accurate quantum computations in less time. As the quality of quantum devices improves, this method will provide a proportional reduction in estimation runtime. This technique may be used to speed up quantum computations into the regime of early fault-tolerant quantum computation and aid in the realization of quantum advantage.
Quantum chemistry and materials is one of the most promising applications of quantum computing. Yet much work is still to be done in matching industry-relevant problems in these areas with quantum algorithms that can solve them. Most previous efforts have carried out resource estimations for quantum algorithms run on large-scale fault-tolerant architectures, which include the quantum phase estimation algorithm. In contrast, few have assessed the performance of near-term quantum algorithms, which include the variational quantum eigensolver (VQE) algorithm. Recently, a large-scale benchmark study [Gonthier et al. 2020] found evidence that the performance of the variational quantum eigensolver for a set of industry-relevant molecules may be too inefficient to be of practical use. This motivates the need for developing and assessing methods that improve the efficiency of VQE. In this work, we predict the runtime of the energy estimation subroutine of VQE when using robust amplitude estimation (RAE) to estimate Pauli expectation values. Under conservative assumptions, our resource estimation predicts that RAE can reduce the runtime over the standard estimation method in VQE by one to two orders of magnitude. Despite this improvement, we find that the runtimes are still too large to be practical. These findings motivate two complementary efforts towards quantum advantage: 1) the investigation of more efficient near-term methods for ground state energy estimation and 2) the development of problem instances that are of industrial value and classically challenging, but better suited to quantum computation.
The number of measurements demanded by hybrid quantum-classical algorithms such as the variational quantum eigensolver (VQE) is prohibitively high for many problems of practical value.For such problems, realizing quantum advantage will require methods that dramatically reduce this cost.Previous quantum algorithms that reduce the measurement cost (e.g., quantum amplitude and phase estimation) require error rates that are too low for near-term implementation.Here we propose methods that take advantage of the available quantum coherence to maximally enhance the power of sampling on noisy quantum devices, reducing the measurement number and runtime compared to the standard sampling method of the VQE.Our scheme derives inspiration from quantum metrology, phase estimation, and the more recent "alpha-VQE" proposal, arriving at a general formulation that is robust to error and does not require ancilla qubits.The central object of this method is what we call the "engineered likelihood function" (ELF), used for carrying out Bayesian inference.We show how the ELF formalism enhances the rate of information gain in sampling as the physical hardware transitions from the regime of noisy intermediate-scale quantum computers to that of quantum error-corrected ones.This technique speeds up a central component of many quantum algorithms, with applications including chemistry, materials, finance, and beyond.Similar to the VQE, we expect small-scale implementations to be realizable on today's quantum devices.
Abstract A universal fault-tolerant quantum computer holds the promise to speed up computational problems that are otherwise intractable on classical computers; however, for the next decade or so, our access is restricted to noisy intermediate-scale quantum (NISQ) computers and, perhaps, early fault tolerant (EFT) quantum computers. This motivates the development of many near-term quantum algorithms including robust amplitude estimation (RAE), which is a quantum-enhanced algorithm for estimating expectation values. One obstacle to using RAE has been a paucity of ways of getting realistic error models incorporated into this algorithm. So far the impact of device noise on RAE is incorporated into one of its subroutines as an exponential decay model, which is unrealistic for NISQ devices and, maybe, for EFT devices; this hinders the performance of RAE. Rather than trying to explicitly model realistic noise effects, which may be infeasible, we circumvent this obstacle by tailoring device noise using randomized compiling to generate an effective noise model, whose impact on RAE closely resembles that of the exponential decay model. Using noisy simulations, we show that our noise-tailored RAE algorithm is able to regain improvements in both bias and precision that are expected for RAE. Additionally, on IBM’s quantum computer ibmq_belem our algorithm demonstrates advantage over the standard estimation technique in reducing bias. Thus, our work extends the feasibility of RAE on NISQ computers, consequently bringing us one step closer towards achieving quantum advantage using these devices.
The Quantum Amplitude Estimation (QAE) algorithm is a major quantum algorithm designed to achieve a quadratic speed-up. Until fault-tolerant quantum computing is achieved, being competitive over classical Monte Carlo (MC) remains elusive. Alternative methods have been developed so as to require fewer resources while maintaining an advantageous theoretical scaling. We compared the standard QAE algorithm with two Noisy Intermediate-Scale Quantum (NISQ)-friendly versions of QAE on a numerical integration task, with the Monte Carlo technique of Metropolis–Hastings as a classical benchmark. The algorithms were evaluated in terms of the estimation error as a function of the number of samples, computational time, and length of the quantum circuits required by the solutions, respectively. The effectiveness of the two QAE alternatives was tested on an 11-qubit trapped-ion quantum computer in order to verify which solution can first provide a speed-up in the integral estimation problems. We concluded that an alternative approach is preferable with respect to employing the phase estimation routine. Indeed, the Maximum Likelihood estimation guaranteed the best trade-off between the length of the quantum circuits and the precision in the integral estimation, as well as greater resistance to noise.
Quantum computers have the potential to deliver speed-ups for solving certain important problems that are intractable for classical counterparts, making them a promising avenue for advancing modern computation. However, many quantum algorithms require deep quantum circuits, which are challenging to implement on current noisy devices. To address this limitation, variational quantum algorithms (VQAs) have been actively developed, enabling practical quantum computing in the noisy intermediate-scale quantum (NISQ) era. Among them, the variational quantum eigensolver (VQE) stands out as a leading approach for solving problems in quantum chemistry, many-body physics, and even integer factorization. The VQE algorithm can be implemented on various quantum hardware platforms, including photonic systems, quantum dots, trapped ions, neutral atoms, and superconducting circuits. In particular, photonic platforms offer several advantages: they operate at room temperature, exhibit low decoherence, and support multiple degrees of freedom, making them suitable for scalable, high-dimensional quantum computation. Here we present methodologies for realizing VQE on photonic systems, highlighting their potential for practical quantum computing. We first provide a theoretical overview of the VQE framework, focusing on the procedure for variationally estimating ground state energies. We then explore how photonic systems can implement these processes, showing that a wide variety of problems can be addressed using either multiple qubit states or a single qudit state.
The number of measurements demanded by hybrid quantum-classical algorithms such as the variational quantum eigensolver (VQE) is prohibitively high for many problems of practical value. Quantum algorithms that reduce this cost (e.g. quantum amplitude and phase estimation) require error rates that are too low for near-term implementation. Here we propose methods that take advantage of the available quantum coherence to maximally enhance the power of sampling on noisy quantum devices, reducing measurement number and runtime compared to VQE. Our scheme derives inspiration from quantum metrology, phase estimation, and the more recent alpha-VQE proposal, arriving at a general formulation that is robust to error and does not require ancilla qubits. The central object of this method is what we call the engineered likelihood function (ELF), used for carrying out Bayesian inference. We show how the ELF formalism enhances the rate of information gain in sampling as the physical hardware transitions from the regime of noisy intermediate-scale quantum computers into that of quantum error corrected ones. This technique speeds up a central component of many quantum algorithms, with applications including chemistry, materials, finance, and beyond. Similar to VQE, we expect small-scale implementations to be realizable on today's quantum devices.