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Quantum algorithm for approximating the expected value of a random-exist quantified oracle

Caleb Rotello

TL;DR

The paper introduces the random-exist quantified oracle (REQO) problem, framing it as $\mu = \mathsf{R} x \exists y \, (\mathbb{E}[f(x,y)])$ and analyzes both classical and quantum approaches to estimate $\mu$ with additive error $\epsilon$. The classical method uses sampling and brute-force search, yielding a query complexity that scales as $O\big((\mu \mathbb{E}_{\lambda_\xi>0}[\lambda_\xi^{-1}] + (1-\mu)2^c)/\epsilon^2\big)$, reflecting the unstructured-search limitation. The quantum algorithm encodes the distribution over $x$ in a quantum state, performs oblivious quantum amplitude amplification to solve per-scenario searches in parallel, and applies quantum amplitude estimation to obtain $\tilde{a}$ at the Heisenberg limit, with total oracle calls scaling as $(L_t+1)(2M-1)$ and a rigorous error bound. The authors prove $\sharp$P-hardness of computing $\mu$ via a reduction from graph reliability and identify a regime (notably $\lambda_\xi = O(2^{-c})$) where the quantum approach achieves a quadratic speedup, $\tilde{O}(\sqrt{2^c}/\epsilon)$, over the classical $\tilde{O}(2^c/\epsilon^2)$, with implications for stochastic programming and uncertainty-aware decision problems.

Abstract

Quantum amplitude amplification and estimation have shown quadratic speedups to unstructured search and estimation tasks. We show that a coherent combination of these quantum algorithms also provides a quadratic speedup to calculating the expectation value of a random-exist quantified oracle. In this problem, Nature makes a decision randomly, i.e. chooses a bitstring according to some probability distribution, and a player has a chance to react by finding a complementary bitstring such that an black-box oracle evaluates to $1$ (or True). Our task is to approximate the probability that the player has a valid reaction to Nature's initial decision. We compare the quantum algorithm to the average-case performance of Monte-Carlo integration over brute-force search, which is, under reasonable assumptions, the best performing classical algorithm. We find the performance separation depends on some problem parameters, and show a regime where the canonical quadratic speedup exists.

Quantum algorithm for approximating the expected value of a random-exist quantified oracle

TL;DR

The paper introduces the random-exist quantified oracle (REQO) problem, framing it as and analyzes both classical and quantum approaches to estimate with additive error . The classical method uses sampling and brute-force search, yielding a query complexity that scales as , reflecting the unstructured-search limitation. The quantum algorithm encodes the distribution over in a quantum state, performs oblivious quantum amplitude amplification to solve per-scenario searches in parallel, and applies quantum amplitude estimation to obtain at the Heisenberg limit, with total oracle calls scaling as and a rigorous error bound. The authors prove P-hardness of computing via a reduction from graph reliability and identify a regime (notably ) where the quantum approach achieves a quadratic speedup, , over the classical , with implications for stochastic programming and uncertainty-aware decision problems.

Abstract

Quantum amplitude amplification and estimation have shown quadratic speedups to unstructured search and estimation tasks. We show that a coherent combination of these quantum algorithms also provides a quadratic speedup to calculating the expectation value of a random-exist quantified oracle. In this problem, Nature makes a decision randomly, i.e. chooses a bitstring according to some probability distribution, and a player has a chance to react by finding a complementary bitstring such that an black-box oracle evaluates to (or True). Our task is to approximate the probability that the player has a valid reaction to Nature's initial decision. We compare the quantum algorithm to the average-case performance of Monte-Carlo integration over brute-force search, which is, under reasonable assumptions, the best performing classical algorithm. We find the performance separation depends on some problem parameters, and show a regime where the canonical quadratic speedup exists.

Paper Structure

This paper contains 8 sections, 3 theorems, 44 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Classical Algorithm
  3. Quantum Algorithm
  4. Oblivious Quantum Amplitude Amplification over a Probability Distribution
  5. Quantum amplitude estimation
  6. Computational complexity of estimating the expectation value
  7. Discussion of advantage
  8. Conclusion

Key Result

Theorem 1

A quantum algorithm exists which can compute an estimate $\tilde{a}$ such that $\text{Pr}[|\tilde{a}-\mu| \leq \epsilon] \geq 8/\pi^2$, where $\epsilon\leq \epsilon_t + \delta^2\mu - \delta^2\epsilon_t +\frac{\pi}{M} + \frac{\pi^2}{M^2}$, with $(L_t+1)(2M-1)$ calls to the oracle $f$.

Figures (3)

  • Figure 1: The dynamics of each $\xi$ subspace's success probability $P_{L,\xi}$ as a function of search depth $l$ (defined as $L=2l+1$, described in detail in Sec \ref{['sec:oqaa']}). We hold $\delta=.3$ constant; each $P_{L,\xi}$, or the orange lines, follows Eq. \ref{['eq:chebyP']}. The horizontal dashed line is at $1-\delta^2$, the convergence floor. The blue line represents the quantity $a$ from Eq. \ref{['eq:a']}, should that $L$ be the chosen amplitude amplification search depth; our search routine is converged once $a$ is within the green region. The vertical dashed line represents $L_t$, or the minimum search depth to converge the search, with $\epsilon_t\leq0.01$. Note that not all of the individual searches have converged at $L_t$, but the quantity we will estimate $a$ has. To generate data: $c=b=6$, with an independent identical distribution for the $x$ register. The oracle $f(\xi,\phi)$ returns 1 if $\xi/8 - 3 > \phi$, with $\xi,\phi$ treated as integers, evaluates to true.
  • Figure 2: The circuit diagram for the algorithm $S_LV$. The first $b$ qubits in the top register host the probability distribution, or randomly-quantified variables, while the last $c$ qubits host the decision or existentially-quantified variables. As we can see, the oracle unitary $U_f$ is operationally identical to the oracle in canonical quantum amplitude amplification algorithms. The mixing unitary only needs to operate on the decision register.
  • Figure 3: The circuit diagram for QAE.

Theorems & Definitions (4)

  • Conjecture 1: Classical approximation algorithm
  • Theorem 1: Oracle query complexity
  • Theorem 2: Oblivious amplitude amplification over a probability distribution
  • Theorem 3: Complexity of computing the expected value of the random-exist quantified oracle