Quantum Computation Relative to Oracles

Atul Singh Arora, Alexandru Gheorghiu, Singh, Uttam

An important theoretical problem in the study of quantum computation, that is also practically relevant in the context of near-term quantum devices, is to understand the computational power of hybrid models, that combine poly-time classical computation with short-depth quantum computation. Here, we consider two such models: CQ_d which captures the scenario of a polynomial-time classical algorithm that queries a d-depth quantum computer many times; and QC_d which is more analogous to measurement-based quantum computation and captures the scenario of a d-depth quantum computer with the ability to change the sequence of gates being applied depending on measurement outcomes processed by a classical computation. Chia, Chung & Lai (STOC 2020) and Coudron & Menda (STOC 2020) showed that these models (with d=log^O(1) (n)) are strictly weaker than BQP (the class of problems solvable by poly-time quantum computation), relative to an oracle, disproving a conjecture of Jozsa in the relativised world. We show that, despite the similarities between CQ_d and QC_d, the two models are incomparable, i.e. CQ_d $\nsubseteq$ QC_d and QC_d $\nsubseteq$ CQ_d relative to an oracle. In other words, there exist problems that one model can solve but not the other and vice versa. We do this by considering new oracle problems that capture the distinctions between the two models and by introducing the notion of an intrinsically stochastic oracle, an oracle whose responses are inherently randomised, which is used for our second result. While we leave showing the second separation relative to a standard oracle as an open problem, we believe the notion of stochastic oracles could be of independent interest for studying complexity classes which have resisted separation in the standard oracle model. Our constructions also yield simpler oracle separations between the hybrid models and BQP, compared to earlier works.

Rafael Pereira da Silva

Stanley Gudder

Matthew Coudron, Sanketh Menda

A conjecture of Jozsa (arXiv:quant-ph/0508124) states that any polynomial-time quantum computation can be simulated by polylogarithmic-depth quantum computation interleaved with polynomial-depth classical computation. Separately, Aaronson conjectured that there exists an oracle O such that BQP O ≠ (BPPBQNC) O . These conjectures are intriguing allusions to the unresolved potential of combining classical and low-depth quantum computation. In this work we show that the Welded Tree Problem, which is an oracle problem that can be solved in quantum polynomial time as shown by Childs et al. (arXiv:quant-ph/0209131), cannot be solved in BPPBQNC, nor can it be solved in the class that Jozsa describes. This proves Aaronson’s oracle separation conjecture and provides a counterpoint to Jozsa’s conjecture relative to the Welded Tree oracle problem. More precisely, we define two complexity classes, HQC and JC whose languages are decided by two different families of interleaved quantum-classical circuits. HQC contains BPPBQNC and is therefore relevant to Aaronson’s conjecture, while JC captures the model of computation that Jozsa considers. We show that the Welded Tree Problem gives an oracle separation between either of {JC, HQC} and BQP. Therefore, even when interleaved with arbitrary polynomial-time classical computation, greater ”quantum depth” leads to strictly greater computational ability in this relativized setting.

Stefano Markidis

Classical-to-quantum I/O is a critical bottleneck for data-intensive quantum workloads. Quantum Oracle Sketching (QOS) has been proposed to address this problem. We identify its practical requirements, including the representation of data as a probability distribution over oracle addresses, the relation between sample count, approximation error, and update cost, and the coherence and runtime support required for online execution. To study QOS on current quantum computing devices, we develop an offline realization that compiles sample blocks into empirical phase oracles and evaluate it using DNA fingerprinting. The application converges with relatively few samples, while exact phase fusion combines repeated updates to the same address. However, the compiled circuits remain too deep for current quantum computing devices. Practical QOS therefore requires lower depth oracle synthesis and interactive quantum runtimes.

Mahn‐Soo Choi