Abstract
We describe an efficient implementation of Bayesian quantum phase estimation in the presence of noise and multiple eigenstates. The main contribution of this work is the dynamic switching between different representations of the phase distributions, namely truncated Fourier series and normal distributions. The Fourier-series representation has the advantage of being exact in many cases, but suffers from increasing complexity with each update of the prior. This necessitates truncation of the series, which eventually causes the distribution to become unstable. We derive bounds on the error in representing normal distributions with a truncated Fourier series, and use these to decide when to switch to the normal-distribution representation. This representation is much simpler, and was proposed in conjunction with rejection filtering for approximate Bayesian updates. We show that, in many cases, the update can be done exactly using analytic expressions, thereby greatly reducing the time complexity of the updates. Finally, when dealing with a superposition of several eigenstates, we need to estimate the relative weights. This can be formulated as a convex optimization problem, which we solve using a gradient-projection algorithm. By updating the weights at exponentially scaled iterations we greatly reduce the computational complexity without affecting the overall accuracy.
Highlights
Phase estimation is an important building block in quantum computing, with applications ranging from ground-state determination in quantum chemistry, to prime factorization and quantum sampling [2, 15, 18]
A similar expression holds for the inner product with cosine terms cos(kφ). We can evaluate these expressions using the Fourier transform of the normal distribution [5, 8], which is given by f(t) =
We compare the performance of the three different approaches: the normal approach in which all priors take the form of a normal distribution; the Fourier approach in which the priors are represented as a truncated Fourier series; and the proposed mixed approach, in which each of the priors is initially represented as a truncated Fourier series and converted to normal distribution form once the standard deviation falls below a given threshold
Summary
Phase estimation is an important building block in quantum computing, with applications ranging from ground-state determination in quantum chemistry, to prime factorization and quantum sampling [2, 15, 18]. It may not be possible to initialize the state exactly to a single eigenstate This could be because the state is perturbed by noise, or because the eigenstate is unknown. Regardless of the cause, phase estimation may need to deal with an initial state that is a superposition of eigenstates:. Practical phase-estimation algorithms may need to deal different sources of noise present in current and near-term quantum devices.
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