© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2302.10332 · Leibniz international proceedings in informatics

A Qubit, a Coin, and an Advice String Walk Into a Relational Problem

Scott Aaronson, Harry Buhrman, William Kretschmer

Published Feb 20, 2023• 6 citations

Abstract

Relational problems (those with many possible valid outputs) are different from decision problems, but it is easy to forget just how different. This paper initiates the study of FBQP/qpoly, the class of relational problems solvable in quantum polynomial-time with the help of polynomial-sized quantum advice, along with its analogues for deterministic and randomized computation (FP, FBPP) and advice (/poly, /rpoly). Our first result is that FBQP/qpoly ≠ FBQP/poly, unconditionally, with no oracle - a striking contrast with what we know about the analogous decision classes. The proof repurposes the separation between quantum and classical one-way communication complexities due to Bar-Yossef, Jayram, and Kerenidis. We discuss how this separation raises the prospect of near-term experiments to demonstrate "quantum information supremacy," a form of quantum supremacy that would not depend on unproved complexity assumptions. Our second result is that FBPP ̸ ⊂ FP/poly - that is, Adleman’s Theorem fails for relational problems - unless PSPACE ⊂ NP/poly. Our proof uses IP = PSPACE and time-bounded Kolmogorov complexity. On the other hand, we show that proving FBPP ̸ ⊂ FP/poly will be hard, as it implies a superpolynomial circuit lower bound for PromiseBPEXP. We prove the following further results: - Unconditionally, FP ≠ FBPP and FP/poly ≠ FBPP/poly (even when these classes are carefully defined). - FBPP/poly = FBPP/rpoly (and likewise for FBQP). For sampling problems, by contrast, SampBPP/poly ≠ SampBPP/rpoly (and likewise for SampBQP).

DOI: 10.4230/lipics.itcs.2024.1

Related papers

Are uncloneable proof and advice states strictly necessary?

Chatterjee, Rohit, Srijita Kundu, Supartha Podder

Yes, we show that they are. We initiate the study of languages that necessarily need uncloneable quantum proofs and advice. We define strictly uncloneable versions of the classes QMA, BQP/qpoly and FEQP/qpoly (which is the class of relational problems solvable exactly with polynomial-sized quantum advice). Strictly uncloneable QMA is defined to be the class of languages in QMA that only have uncloneable proofs, i.e., given any family of candidate proof states, a polynomial-time cloning algorithm cannot act on it to produce states that are jointly usable by $k$ separate polynomial-time verifiers, for arbitrary polynomial $k$. This is a stronger notion of uncloneable proofs and advice than those considered in previous works, which only required the existence of a single family of proof or advice states that are uncloneable. We show that in the quantum oracle model, there exist languages in strictly uncloneable QMA and strictly uncloneable BQP/qpoly. The language in strictly uncloneable QMA also gives a quantum oracle separation between QMA and the class cloneableQMA introduced by Nehoran and Zhandry (2024). We also show without using any oracles that the language, used by Aaronson, Buhrman and Kretschmer (2024) to separate FEQP/qpoly and FBQP/poly, is in strictly uncloneable FEQP/qpoly.

arXiv:2410.11827•Oct 15, 2024•0 citations

Complexity-Theoretic Foundations of Quantum Supremacy Experiments

Scott Aaronson, Lijie Chen

Depending on time, I'll discuss some subset of the following results. (1) There exists an oracle relative to which classical and quantum computers can solve the same approximate sampling problems in polynomial time, and yet the polynomial hierarchy is infinite. This means that any "BosonSampling-type theorem" will need to be non-relativizing. (2) Can P be separated from BQP by an efficiently-computable oracle (one in P/poly)? Yes if quantum-resistant one-way functions exist; no if P=PSPACE. The latter builds on recent results in quantum query complexity by myself and Shalev Ben-David. (3) How hard is it to approximately sample the output distribution of a random quantum circuit? One can do it (and even calculate the probabilities) in polynomial space and m^O(n) time, where m is the number of gates and n is the number of qubits. But if that's essentially optimal for estimating probabilities, then approximate sampling also requires exponential time. Joint work with Lijie Chen (Tsinghua University)

arXiv:1612.05903•Leibniz international proceedings in informatics•Dec 18, 2016•218 citations

Quantum Meta-Complexity Is All You Need: Characterizing One-Way Puzzles via Time-Bounded Kolmogorov Complexity

Morteza Saberikamarposhti

We initiate the time-bounded meta-complexity program for quantum cryptography. Recent work characterizes one-way puzzles, the minimal search primitive of quantum cryptography without one-way functions, by the average-case hardness of approximating the plain, uncomputable Kolmogorov complexity over quantumly samplable distributions; the classical program of Liu and Pass, by contrast, lives at polynomial time bounds. We define a probabilistic time-bounded quantum program complexity pKq^t for classical strings and prove two unconditional theorems. First, a quantum coding theorem: any string output by a quantum polynomial-time sampler with probability delta admits a description of the information-theoretically optimal length log(1/delta) plus logarithmic terms, decodable by a quantum machine in time O(sqrt(1/delta)) times a polynomial, via amplitude amplification over the coherently executed sampler. Second, an exact characterization at subexponential time: one-way puzzles exist if and only if the gap problem for pKq at time bound 2^(n/2) poly(n) is weakly quantum-average-hard, refining the plain-complexity characterizations. We then isolate the polynomial-time coding theorem as the single load-bearing open conjecture of the program, prove that it implies the full polynomial-time characterization, analyze why the classical derandomization proof resists quantization, and formulate a relativized barrier conjecture delimiting string-valued meta-complexity at one-way puzzles. Conjectures are labeled as such throughout.

quant-pharXiv:2609.02687•Sep 3, 2026•0 citations

Classical vs Quantum Advice and Proofs Under Classically-Accessible Oracle

Xingjian Li, Liu, Qipeng, Angelos Pelecanos +1 more

It is a long-standing open question to construct a classical oracle relative to which BQP/qpoly $\neq$ BQP/poly or QMA $\neq$ QCMA. In this paper, we construct classically-accessible classical oracles relative to which BQP/qpoly $\neq$ BQP/poly and QMA $\neq$ QCMA. Here, classically-accessible classical oracles are oracles that can be accessed only classically even for quantum algorithms. Based on a similar technique, we also show an alternative proof for the separation of QMA and QCMA relative to a distributional quantumly-accessible classical oracle, which was recently shown by Natarajan and Nirkhe.

arXiv:2303.04298•Leibniz international proceedings in informatics•Mar 8, 2023•2 citations

No Size-Preserving Amplification with Quantum Advice

Shih-Han Hung, Han-Hsuan Lin

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 $m(n)=ω(1)$ and every error bound $\varepsilon(n)$ that stays below $1/2$ by at least an inverse polynomial, there is a positive function $δ$ with $δ(n)=O\bigl(\min\{(\log m/m)^{1/4},\ \sqrt{\log m/m}\,/(1/2-\varepsilon(n))\}\bigr)$ such that $\mathsf{BQP}_{\varepsilon}/\mathsf{q}m \subsetneq \mathsf{BQP}_{\varepsilon + δ}/\mathsf{q}m$; for constant $\varepsilon$ the gap is $O(\sqrt{\log m/m})$. Here, $\mathsf{BQP}_\varepsilon/\mathsf{q}m$ is the class of languages recognizable with error at most $\varepsilon(n)$ by a polynomial-time quantum algorithm with an $m(n)$-qubit advice state that only depends on the input length $n$. We show this by proving a stronger separation $\mathsf{P}_{\varepsilon+δ}/\mathsf{r} m \not\subset \mathsf{BQP}_{\varepsilon}/\mathsf{q} m$, where $\mathsf{P}_{\varepsilon}/\mathsf{r}m$ is the class of languages recognizable with error at most $\varepsilon(n)$ by a deterministic polynomial-time algorithm with an $m(n)$-bit advice string sampled from a distribution that depends only on $n$.

quant-pharXiv:2610.03377•Oct 5, 2026•0 citations

PDQP/qpoly=ALL

Scott Aaronson

We show that combining two different hypothetical enhancements to quantum computation---namely, quantum advice and non-collapsing measurements---would let a quantum computer solve any decision problem whatsoever in polynomial time, even though neither enhancement yields extravagant power by itself. This complements a related result due to Raz. The proof uses locally decodable codes.

Quantum Information and Computation•Sep 1, 2018•3 citations

Relational problems (those with many possible valid outputs) are different from decision problems, but it is easy to forget just how different. This paper initiates the study of FBQP/qpoly, the class of relational problems solvable in quantum polynomial-time with the help of polynomial-sized quantum advice, along with its analogues for deterministic and randomized computation (FP, FBPP) and advice (/poly, /rpoly). Our first result is that FBQP/qpoly ≠ FBQP/poly, unconditionally, with no oracle - a striking contrast with what we know about the analogous decision classes. The proof repurposes the separation between quantum and classical one-way communication complexities due to Bar-Yossef, Jayram, and Kerenidis. We discuss how this separation raises the prospect of near-term experiments to demonstrate "quantum information supremacy," a form of quantum supremacy that would not depend on unproved complexity assumptions. Our second result is that FBPP ̸ ⊂ FP/poly - that is, Adleman’s Theorem fails for relational problems - unless PSPACE ⊂ NP/poly. Our proof uses IP = PSPACE and time-bounded Kolmogorov complexity. On the other hand, we show that proving FBPP ̸ ⊂ FP/poly will be hard, as it implies a superpolynomial circuit lower bound for PromiseBPEXP. We prove the following further results: - Unconditionally, FP ≠ FBPP and FP/poly ≠ FBPP/poly (even when these classes are carefully defined). - FBPP/poly = FBPP/rpoly (and likewise for FBQP). For sampling problems, by contrast, SampBPP/poly ≠ SampBPP/rpoly (and likewise for SampBQP).

Yes, we show that they are. We initiate the study of languages that necessarily need uncloneable quantum proofs and advice. We define strictly uncloneable versions of the classes QMA, BQP/qpoly and FEQP/qpoly (which is the class of relational problems solvable exactly with polynomial-sized quantum advice). Strictly uncloneable QMA is defined to be the class of languages in QMA that only have uncloneable proofs, i.e., given any family of candidate proof states, a polynomial-time cloning algorithm cannot act on it to produce states that are jointly usable by

k

separate polynomial-time verifiers, for arbitrary polynomial

k

. This is a stronger notion of uncloneable proofs and advice than those considered in previous works, which only required the existence of a single family of proof or advice states that are uncloneable. We show that in the quantum oracle model, there exist languages in strictly uncloneable QMA and strictly uncloneable BQP/qpoly. The language in strictly uncloneable QMA also gives a quantum oracle separation between QMA and the class cloneableQMA introduced by Nehoran and Zhandry (2024). We also show without using any oracles that the language, used by Aaronson, Buhrman and Kretschmer (2024) to separate FEQP/qpoly and FBQP/poly, is in strictly uncloneable FEQP/qpoly.

Depending on time, I'll discuss some subset of the following results. (1) There exists an oracle relative to which classical and quantum computers can solve the same approximate sampling problems in polynomial time, and yet the polynomial hierarchy is infinite. This means that any "BosonSampling-type theorem" will need to be non-relativizing. (2) Can P be separated from BQP by an efficiently-computable oracle (one in P/poly)? Yes if quantum-resistant one-way functions exist; no if P=PSPACE. The latter builds on recent results in quantum query complexity by myself and Shalev Ben-David. (3) How hard is it to approximately sample the output distribution of a random quantum circuit? One can do it (and even calculate the probabilities) in polynomial space and m^O(n) time, where m is the number of gates and n is the number of qubits. But if that's essentially optimal for estimating probabilities, then approximate sampling also requires exponential time. Joint work with Lijie Chen (Tsinghua University)

We initiate the time-bounded meta-complexity program for quantum cryptography. Recent work characterizes one-way puzzles, the minimal search primitive of quantum cryptography without one-way functions, by the average-case hardness of approximating the plain, uncomputable Kolmogorov complexity over quantumly samplable distributions; the classical program of Liu and Pass, by contrast, lives at polynomial time bounds. We define a probabilistic time-bounded quantum program complexity pKq^t for classical strings and prove two unconditional theorems. First, a quantum coding theorem: any string output by a quantum polynomial-time sampler with probability delta admits a description of the information-theoretically optimal length log(1/delta) plus logarithmic terms, decodable by a quantum machine in time O(sqrt(1/delta)) times a polynomial, via amplitude amplification over the coherently executed sampler. Second, an exact characterization at subexponential time: one-way puzzles exist if and only if the gap problem for pKq at time bound 2^(n/2) poly(n) is weakly quantum-average-hard, refining the plain-complexity characterizations. We then isolate the polynomial-time coding theorem as the single load-bearing open conjecture of the program, prove that it implies the full polynomial-time characterization, analyze why the classical derandomization proof resists quantization, and formulate a relativized barrier conjecture delimiting string-valued meta-complexity at one-way puzzles. Conjectures are labeled as such throughout.

It is a long-standing open question to construct a classical oracle relative to which BQP/qpoly

\neq

BQP/poly or QMA

\neq

QCMA. In this paper, we construct classically-accessible classical oracles relative to which BQP/qpoly

\neq

BQP/poly and QMA

\neq

QCMA. Here, classically-accessible classical oracles are oracles that can be accessed only classically even for quantum algorithms. Based on a similar technique, we also show an alternative proof for the separation of QMA and QCMA relative to a distributional quantumly-accessible classical oracle, which was recently shown by Natarajan and Nirkhe.

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)

and every error bound

\varepsilon(n)

that stays below

1/2

by at least an inverse polynomial, there is a positive function

δ

with

δ(n)=O\bigl(\min\{(\log m/m)^{1/4},\ \sqrt{\log m/m}\,/(1/2-\varepsilon(n))\}\bigr)

such that

\mathsf{BQP}_{\varepsilon}/\mathsf{q}m \subsetneq \mathsf{BQP}_{\varepsilon + δ}/\mathsf{q}m

; for constant

\varepsilon

the gap is

O(\sqrt{\log m/m})

. Here,

\mathsf{BQP}_\varepsilon/\mathsf{q}m

is the class of languages recognizable with error at most

\varepsilon(n)

by a polynomial-time quantum algorithm with an

m(n)

-qubit advice state that only depends on the input length

n

. We show this by proving a stronger separation

\mathsf{P}_{\varepsilon+δ}/\mathsf{r} m \not\subset \mathsf{BQP}_{\varepsilon}/\mathsf{q} m

, where

\mathsf{P}_{\varepsilon}/\mathsf{r}m

is the class of languages recognizable with error at most

\varepsilon(n)

by a deterministic polynomial-time algorithm with an

m(n)

-bit advice string sampled from a distribution that depends only on

n

.