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arXiv:2403.09994 · Theory of Computing

Even quantum advice is unlikely to solve PP

Justin Yirka

Published Jan 1, 2025• 1 citation
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Abstract

We give a corrected proof that if $\mathsf{PP} \subseteq \mathsf{BQP/qpoly}$ (probabilistic polynomial time can be efficiently simulated by quantum circuits with quantum advice), then the Counting Hierarchy collapses, as originally claimed by Aaronson (CCC'06). This recovers the related unconditional claim that $\mathsf{PP}$ does not have circuits of any fixed-polynomial size $n^k$ even with quantum advice. Our result is based on proving that $\mathsf{YQP^*}$, an oblivious version of $\mathsf{QMA}\cap\mathsf{coQMA}$, is contained in $\mathsf{APP}$, a $\mathsf{PP}$-low subclass of $\mathsf{PP}$ with an arbitrarily small but nonzero promise gap.

DOI
10.4086/toc.2025.v021a007

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We give a corrected proof that if PP⊆BQP/qpoly\mathsf{PP} \subseteq \mathsf{BQP/qpoly}PP⊆BQP/qpoly (probabilistic polynomial time can be efficiently simulated by quantum circuits with quantum advice), then the Counting Hierarchy collapses, as originally claimed by Aaronson (CCC'06). This recovers the related unconditional claim that PP\mathsf{PP}PP does not have circuits of any fixed-polynomial size nkn^knk even with quantum advice. Our result is based on proving that YQP∗\mathsf{YQP^*}YQP∗, an oblivious version of QMA∩coQMA\mathsf{QMA}\cap\mathsf{coQMA}QMA∩coQMA, is contained in APP\mathsf{APP}APP, a PP\mathsf{PP}PP-low subclass of PP\mathsf{PP}PP with an arbitrarily small but nonzero promise gap.
We prove the following surprising result: given any quantum state rho on n qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of H can be used to simulate rho on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly is contained in PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in power to untrusted quantum advice combined with trusted classical advice.
We prove the following surprising result: given any quantum state ρ\rhoρ on nnn qubits, there exists a local Hamiltonian HHH on poly⁡(n)\operatorname*{poly}(n)poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of HHH can be used to simulate ρ\rhoρ on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly⊆QMA/poly{BQP/qpoly}\subseteq{QMA/poly}BQP/qpoly⊆QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly⊆PP/poly{BQP/qpoly}\subseteq {PP/poly}BQP/qpoly⊆PP/poly. Indeed, we can exactly characterize quantum advice as equivalent in power to untrusted quantum advice combined with trusted classical advice. Proving our main result requires combining a large number of previous tools---including a result of Alon et al. on learning of real-valued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on “QMA+{QMA}_{{+}}QMA+​ super-verifiers''---and also creating some new ones. The main new tool is a so-called majority-certificates lemma, which is closely related to boosting in machine learning, and which seems likely to find independent applications. In its simplest version, this lemma says the following. Given any set SSS of Boolean functions on nnn variables, any function f∈Sf\in Sf∈S can be expressed as the pointwise majority of m=O(n)m=O(n)m=O(n) functions f1,…,fm∈Sf_{1},\ldots,f_{m}\in Sf1​,…,fm​∈S, such that each fif_{i}fi​ is the unique function in SSS compatible with O(log⁡∣S∣)O(\log\vert S\vert)O(log∣S∣) input/output constraints.
We prove the following surprising result: given any quantum state rho on n qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of H can be used to simulate rho on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly is contained in PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in power to untrusted quantum advice combined with trusted classical advice. Proving our main result requires combining a large number of previous tools -- including a result of Alon et al. on learning of real-valued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new ones. The main new tool is a so-called majority-certificates lemma, which is closely related to boosting in machine learning, and which seems likely to find independent applications. In its simplest version, this lemma says the following. Given any set S of Boolean functions on n variables, any function f in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm in S, such that each fi is the unique function in S compatible with O(log|S|) input/output constraints.
Marriott and Watrous showed that quantum Merlin--Arthur games admit generic error reduction without increasing witness size [Computational Complexity, 2005]. In this work, we show that this state-size-preserving amplification property does not hold for polynomial-time quantum computation with quantum advice. In particular, we present decision problems for which even a vanishing additive error reduction requires longer advice. More precisely, for every polynomially bounded advice length m(n)=ω(1)m(n)=ω(1)m(n)=ω(1) and every error bound ε(n)\varepsilon(n)ε(n) that stays below 1/21/21/2 by at least an inverse polynomial, there is a positive function δδδ with δ(n)=O(min⁡{(log⁡m/m)1/4, log⁡m/m /(1/2−ε(n))})δ(n)=O\bigl(\min\{(\log m/m)^{1/4},\ \sqrt{\log m/m}\,/(1/2-\varepsilon(n))\}\bigr)δ(n)=O(min{(logm/m)1/4, logm/m​/(1/2−ε(n))}) such that BQPε/qm⊊BQPε+δ/qm\mathsf{BQP}_{\varepsilon}/\mathsf{q}m \subsetneq \mathsf{BQP}_{\varepsilon + δ}/\mathsf{q}mBQPε​/qm⊊BQPε+δ​/qm; for constant ε\varepsilonε the gap is O(log⁡m/m)O(\sqrt{\log m/m})O(logm/m​). Here, BQPε/qm\mathsf{BQP}_\varepsilon/\mathsf{q}mBQPε​/qm is the class of languages recognizable with error at most ε(n)\varepsilon(n)ε(n) by a polynomial-time quantum algorithm with an m(n)m(n)m(n)-qubit advice state that only depends on the input length nnn. We show this by proving a stronger separation Pε+δ/rm⊄BQPε/qm\mathsf{P}_{\varepsilon+δ}/\mathsf{r} m \not\subset \mathsf{BQP}_{\varepsilon}/\mathsf{q} mPε+δ​/rm⊂BQPε​/qm, where Pε/rm\mathsf{P}_{\varepsilon}/\mathsf{r}mPε​/rm is the class of languages recognizable with error at most ε(n)\varepsilon(n)ε(n) by a deterministic polynomial-time algorithm with an m(n)m(n)m(n)-bit advice string sampled from a distribution that depends only on nnn.