Abstract
We consider performing phase estimation under the following conditions: we are given only one copy of the input state, the input state does not have to be an eigenstate of the unitary, and the state must not be measured. Most quantum estimation algorithms make assumptions that make them unsuitable for this 'coherent' setting, leaving only the textbook approach. We present novel algorithms for phase, energy, and amplitude estimation that are both conceptually and computationally simpler than the textbook method, featuring both a smaller query complexity and ancilla footprint. They do not require a quantum Fourier transform, and they do not require a quantum sorting network to compute the median of several estimates. Instead, they use block-encoding techniques to compute the estimate one bit at a time, performing all amplification via singular value transformation. These improved subroutines accelerate the performance of quantum Metropolis sampling and quantum Bayesian inference.
Highlights
We consider performing phase estimation under the following conditions: we are given only one copy of the input state, the input state does not have to be an eigenstate of the unitary, and the state must not be measured
Phase estimation only delivers a quadratic speedup for estimation and the exponential speedup for linear algebra can sidestep the Quantum Fourier Transform (QFT) [CKS15, GSLW18]
Phase estimation requires Hamiltonian simulation, a quantum subroutine that is an entire subject of study in its own right: it requires recent innovations to apply optimally in a black-box setting [BCK15, LC1606, LC1610, LC17, GSLW18] and optimal Hamiltonian simulation for specific systems is still being actively studied [SHC20, Cam20]
Summary
We formally define the estimation tasks we want to solve (Definition 2). This means that the amplitude cannot be 0 or 1 everywhere, there must exist points where it crosses intermediate values in order to interpolate in between the two In both the ‘n-bit estimate’ case and the additive-error case, this causes a problem for coherent quantum algorithms since the estimate cannot always be uncomputed. Median amplification alone is not sufficient to accomplish phase estimation as we have defined it in Definition 1 This is because any λi that is ≈ 10% · 2−n−1 close to a multiple of 1/2n, the probability of correctly obtaining floor(2nλj) is less than 1/2!. For some probabilities pj,x and phases φj,x, since this is just a general description of a unitary map that leaves the |ψj register intact This way of writing the map lets us bound the error in diamond norm to the ideal map. We can conclude that pj,k ≥ 1 − δmed, so the modified algorithm retains the same accuracy
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