Amplitude estimation without phase estimation

Abstract

This paper focuses on the quantum amplitude estimation algorithm, which is a core subroutine in quantum computation for various applications. The conventional approach for amplitude estimation is to use the phase estimation algorithm, which consists of many controlled amplification operations followed by a quantum Fourier transform. However, the whole procedure is hard to implement with current and near-term quantum computers. In this paper, we propose a quantum amplitude estimation algorithm without the use of expensive controlled operations; the key idea is to utilize the maximum likelihood estimation based on the combined measurement data produced from quantum circuits with different numbers of amplitude amplification operations. Numerical simulations we conducted demonstrate that our algorithm asymptotically achieves nearly the optimal quantum speedup with a reasonable circuit length.Quantum computers are expected to allow us to perform high-speed computations over classical computations for problems in a wide range of scientific and technological fields. Environments in which quantum algorithms can be executed by real quantum devices are currently being provided [1][2][3]. Real quantum devices with several tens of qubits will soon be realized in near future, although those are the socalled noisy intermediate-scale quantum (NISQ) devices that impose several practical limitations on their Y. Suzuki, S. Uno: Equally contributing authors.

Option Pricing using Quantum Computers

“…However, research to improve the algorithms is ongoing [52][53][54]. Here we have used a new algorithm [22] that retains the AE speed-up but that uses less gates to measure the price of an option. Furthermore, we employed a simple error mitigation scheme that allowed us to greatly reduce the errors from the noisy quantum hardware.…”

“…To reduce the required number of qubits and the resulting circuit depth, Suzuki et al have shown that AE can be performed without requiring quantum phase estimation while still maintaining a quadratic speed-up [22]. To this extent, they exploit that…”

“…Eq. (22) shows that by setting g 0 = π/4 − c, we could recover E[max(0, i−K)] from P 1 up to a scaling factor and a constant, from which we can subsequently recover the expected payoff E[f (i)] of the option given the probability distribution of the underlying asset. We should note that the fair value of the option requires appropriately discounting the expected payoff of the option to today, but as the discounting can be performed after the expectation value has been calculated, we omit it from our discussion for simplicity.…”

“…In the remainder of this section, we focus on QA |0 3 , i.e., the outlined algorithm for m = 1, to examine the reach of today's quantum hardware in evaluating AE option pricing circuits which do not require phase estimation. We note that this evaluation is relevant to any quantum algorithm realizing AE without phase estimation and is independent of the approach described in [22] or any other particular implementation.…”

“…5 we show results from eval-uating our option circuits on quantum hardware, and describe the error mitigation scheme we employ to increase the accuracy of the estimated option prices. In particular, we employ the maximum likelihood estimation method introduced in [22] to perform amplitude estimation without phase estimation in option pricing using three qubits of a real quantum device.…”