Separating QMA from QCMA with a classical oracle

TL;DR

This work presents a relativized separation between QMA and QCMA by constructing a classical oracle based on Spectral Forrelation. The authors show that a quantum verifier with a quantum witness can decisively solve Spectral Forrelation instances that resist efficient QCMA verification using a classical witness, by reducing QCMA lower bounds to sampling hardness and then proving strong sampling upper bounds via a novel bosonic, compressed-oracle framework. The core technical innovation is a second-quantization perspective that encodes a random multiset S as a bosonic system, enabling precise control over how oracle queries to U affect post-query states and sampling probabilities. By proving that polynomial-query QCMA algorithms would imply too-good samplers for sampling from S, while quantum witnesses cannot, they derive a classical oracle separation between QMA and QCMA. The results connect to broader themes in quantum query complexity, pseudorandomness, and the two-basis paradigm, and introduce a potentially broadly applicable bosonic approach to compressed-oracle techniques.

Abstract

We construct a classical oracle proving that, in a relativized setting, the set of languages decidable by an efficient quantum verifier with a quantum witness (QMA) is strictly bigger than those decidable with access only to a classical witness (QCMA). The separating classical oracle we construct is for a decision problem we coin spectral Forrelation -- the oracle describes two subsets of the boolean hypercube, and the computational task is to decide if there exists a quantum state whose standard basis measurement distribution is well supported on one subset while its Fourier basis measurement distribution is well supported on the other subset. This is equivalent to estimating the spectral norm of a "Forrelation" matrix between two sets that are accessible through membership queries. Our lower bound derives from a simple observation that a query algorithm with a classical witness can be run multiple times to generate many samples from a distribution, while a quantum witness is a "use once" object. This observation allows us to reduce proving a QCMA lower bound to proving a sampling hardness result which does not simultaneously prove a QMA lower bound. To prove said sampling hardness result for QCMA, we observe that quantum access to the oracle can be compressed by expressing the problem in terms of bosons -- a novel "second quantization" perspective on compressed oracle techniques, which may be of independent interest. Using this compressed perspective on the sampling problem, we prove the sampling hardness result, completing the proof.

Figures (5)

• Figure 1: A cartoon of spectral Forrelation. The subsets $S$ and $U$ occupy support in the standard/position and Hadamard/momentum bases, respectively. The sets $S$ and $U$ are $\ge \alpha$-spectrally Forrelated if there exists a state $\ket{\psi}$ such that $\norm{\Pi_U H^{\otimes n} \Pi_S \ket{\psi}}^2 \ge \alpha$. Equivalently, there exists a state $\ket{\psi}$ for which the induced classical distributions by measuring in the standard and Hadamard bases are well supported on $S$ and $U$, respectively.
• Figure 2: Adaptive sampling from a classical witness. A quantum witness $|\psi\rangle$ yields a single sample $x$ upon measurement with an algorithm accessing only $U$. Whereas, a classical witness $w$ can be reused across successive sampling rounds with an adaptive sampler accessing only $U$ and prior samples to generate multiple distinct samples.
• Figure 3: A depiction of how the pair $(S,U)$ is sampled: (a) First, the multiset $S$ is first sampled uniformly randomly; (b) second, for each $y \in \{0,1\}^n$, a parameter $\gamma_y^{(S)}$ is calculated; (c) third, a set $U$ is sampled by including $y$ with independent probability $1 - \frac{1}{2} \mathrm{e}^{-\kappa \gamma_y^{(S)}}$.
• Figure 4: A depiction of the bosonic Fock state $\ket{\psi} = \ket{10, 0, 2, 0, 2, 0, \ldots, 1}$.
• Figure 5: A depiction of $\widetilde{a}_{0 \oplus 3}^\dagger \widetilde{a}_{0 \oplus 3}^\dagger \widetilde{a}_0 \widetilde{a}_0 \ket{\psi}$ which is one component in the sum $\widetilde{\mathbf{H}}_3 \ket{\psi}$.

Theorems & Definitions (93)

• Theorem 1.1: Classical oracle separation between $\QMA$ and $\QCMA$
• Theorem 1.2: Good samplers from $\QCMA$ algorithms (informal)
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Theorem 1.6: Sampling probability upper bound (informal)
• Definition 5.1: Quantum query circuit/algorithm
• Definition 5.2: Quantum query algorithm with witness
• Definition 5.3: Quantum oracle circuits
• Definition 5.4: Oracle $\BQP$