BQP and PPAD

Yang D. Li

Published Aug 1, 2011• 0 citations

Abstract

We initiate the study of the relationship between two complexity classes, BQP (Bounded-Error Quantum Polynomial-Time) and PPAD (Polynomial Parity Argument, Directed). We first give a conjecture that PPAD is contained in BQP, and show a necessary and sufficient condition for the conjecture to hold. Then we prove that the conjecture is not true under the oracle model. In the end, we raise some interesting open problems/future directions.

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BQP and PPAD

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OpenAlex

Abstract

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The relationship between BQP and PH has been an open problem since the earliest days of quantum computing. We present evidence that quantum computers can solve problems outside the entire polynomial hierarchy, by relating this question to topics in circuit complexity, pseudorandomness, and Fourier analysis. First, we show that there exists an oracle relation problem (i.e., a problem with many valid outputs) that is solvable in BQP, but not in PH. This also yields a non-oracle relation problem that is solvable in quantum logarithmic time, but not in AC0. Second, we show that an oracle decision problem separating BQP from PH would follow from the Generalized Linial-Nisan Conjecture, which we formulate here and which is likely of independent interest. The original Linial-Nisan Conjecture (about pseudorandomness against constant-depth circuits) was recently proved by Braverman, after being open for twenty years.The relationship between BQP and PH has been an open problem since the earliest days of quantum computing. We present evidence that quantum computers can solve problems outside the entire polynomial hierarchy, by relating this question to topics in circuit complexity, pseudorandomness, and Fourier analysis. First, we show that there exists an oracle relation problem (i.e., a problem with many valid outputs) that is solvable in BQP, but not in PH. This also yields a non-oracle relation problem that is solvable in quantum logarithmic time, but not in AC0. Second, we show that an oracle decision problem separating BQP from PH would follow from the Generalized Linial-Nisan Conjecture, which we formulate here and which is likely of independent interest. The original Linial-Nisan Conjecture (about pseudorandomness against constant-depth circuits) was recently proved by Braverman, after being open for twenty years.

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One can fix the randomness used by a randomized algorithm, but there is no analogous notion of fixing the quantumness used by a quantum algorithm. Underscoring this fundamental difference, we show that, in the black-box setting, the behavior of quantum polynomial-time ( $\mathsf{BQP}$ ) can be remarkably decoupled from that of classical complexity classes like $\mathsf{NP}$ . Specifically: -There exists an oracle relative to which $\mathsf{NP^{BQP}}\not\subset\mathsf{BQP^{PH}}$ , resolving a 2005 problem of Fortnow. As a corollary, there exists an oracle relative to which $\mathsf{P}=\mathsf{NP}$ but $\mathsf{BQP}\neq\mathsf{QCMA}$ . -Conversely, there exists an oracle relative to which $\mathsf{BQP^{NP}}\not\subset\mathsf{PH^{BQP}}$ . -Relative to a random oracle, $\mathsf{PP}=\mathsf{PostBQP}$ is not contained in the " $\mathsf{QMA}$ hierarchy" $\mathsf{QMA}^{\mathsf{QMA}^{\mathsf{QMA}^{\cdots}}}$ . -Relative to a random oracle, $\mathsfΣ_{k+1}^\mathsf{P}\not\subset\mathsf{BQP}^{\mathsfΣ_{k}^\mathsf{P}}$ for every $k$ . -There exists an oracle relative to which $\mathsf{BQP}=\mathsf{P^{\# P}}$ and yet $\mathsf{PH}$ is infinite. -There exists an oracle relative to which $\mathsf{P}=\mathsf{NP}\neq\mathsf{BQP}=\mathsf{P^{\# P}}$ . To achieve these results, we build on the 2018 achievement by Raz and Tal of an oracle relative to which $\mathsf{BQP}\not \subset \mathsf{PH}$ , and associated results about the Forrelation problem. We also introduce new tools that might be of independent interest. These include a "quantum-aware" version of the random restriction method, a concentration theorem for the block sensitivity of $\mathsf{AC^0}$ circuits, and a (provable) analogue of the Aaronson-Ambainis Conjecture for sparse oracles.One can fix the randomness used by a randomized algorithm, but there is no analogous notion of fixing the quantumness used by a quantum algorithm. Underscoring this fundamental difference, we show that, in the black-box setting, the behavior of quantum polynomial-time ($\mathsf{BQP}$) can be remarkably decoupled from that of classical complexity classes like $\mathsf{NP}$. Specifically: -There exists an oracle relative to which $\mathsf{NP^{BQP}}\not\subset\mathsf{BQP^{PH}}$, resolving a 2005 problem of Fortnow. As a corollary, there exists an oracle relative to which $\mathsf{P}=\mathsf{NP}$ but $\mathsf{BQP}\neq\mathsf{QCMA}$. -Conversely, there exists an oracle relative to which $\mathsf{BQP^{NP}}\not\subset\mathsf{PH^{BQP}}$. -Relative to a random oracle, $\mathsf{PP}=\mathsf{PostBQP}$ is not contained in the "$\mathsf{QMA}$ hierarchy" $\mathsf{QMA}^{\mathsf{QMA}^{\mathsf{QMA}^{\cdots}}}$. -Relative to a random oracle, $\mathsfΣ_{k+1}^\mathsf{P}\not\subset\mathsf{BQP}^{\mathsfΣ_{k}^\mathsf{P}}$ for every $k$. -There exists an oracle relative to which $\mathsf{BQP}=\mathsf{P^{\# P}}$ and yet $\mathsf{PH}$ is infinite. -There exists an oracle relative to which $\mathsf{P}=\mathsf{NP}\neq\mathsf{BQP}=\mathsf{P^{\# P}}$. To achieve these results, we build on the 2018 achievement by Raz and Tal of an oracle relative to which $\mathsf{BQP}\not \subset \mathsf{PH}$, and associated results about the Forrelation problem. We also introduce new tools that might be of independent interest. These include a "quantum-aware" version of the random restriction method, a concentration theorem for the block sensitivity of $\mathsf{AC^0}$ circuits, and a (provable) analogue of the Aaronson-Ambainis Conjecture for sparse oracles.

arXiv:2111.10409Leibniz international proceedings in informaticsNov 19, 20217 citations

A Relativizing MIP for BQP

Scott Aaronson, Anand Natarajan, Avishay Tal +1 more

Complexity class containments involving interactive proof classes are famously nonrelativizing: although $\mathsf{IP} = \mathsf{PSPACE}$ , Fortnow and Sipser showed that that there exists an oracle relative to which $\mathsf{coNP} \not\subseteq \mathsf{IP}$ . In contrast, the question of whether the containment $\mathsf{BQP} \subseteq \mathsf{IP}$ is relativizing remains wide open. In this work we make progress towards resolving this question by showing that the containment $\mathsf{BQP} \subseteq \mathsf{MIP}$ holds with respect to any classical oracle. We obtain this result by constructing, for any classical oracle $O$ , a $\mathsf{PCP}$ proof system for $\mathsf{BQP}^{O}$ where the verifier makes polynomially many classical queries to an exponentially-long proof, and to the oracle $O$ . Our construction is inspired by the state synthesis algorithm of Grover and Rudolph, and serves as a complement to the "exponential PCP" constructed by Aharonov, Arad, and Vidick, which achieves similar parameters but which is based on different ideas and does not relativize. We propose relativization as a proxy for prover efficiency, and hope that progress towards an $\mathsf{IP}$ for $\mathsf{BQP}$ in the oracle world will lead to a non-cryptographic interactive protocol for proving any quantum computation to a classical skeptic in the unrelativized world, which is a longstanding open problem in quantum complexity theory.Complexity class containments involving interactive proof classes are famously nonrelativizing: although $\mathsf{IP} = \mathsf{PSPACE}$, Fortnow and Sipser showed that that there exists an oracle relative to which $\mathsf{coNP} \not\subseteq \mathsf{IP}$. In contrast, the question of whether the containment $\mathsf{BQP} \subseteq \mathsf{IP}$ is relativizing remains wide open. In this work we make progress towards resolving this question by showing that the containment $\mathsf{BQP} \subseteq \mathsf{MIP}$ holds with respect to any classical oracle. We obtain this result by constructing, for any classical oracle $O$, a $\mathsf{PCP}$ proof system for $\mathsf{BQP}^{O}$ where the verifier makes polynomially many classical queries to an exponentially-long proof, and to the oracle $O$. Our construction is inspired by the state synthesis algorithm of Grover and Rudolph, and serves as a complement to the "exponential PCP" constructed by Aharonov, Arad, and Vidick, which achieves similar parameters but which is based on different ideas and does not relativize. We propose relativization as a proxy for prover efficiency, and hope that progress towards an $\mathsf{IP}$ for $\mathsf{BQP}$ in the oracle world will lead to a non-cryptographic interactive protocol for proving any quantum computation to a classical skeptic in the unrelativized world, which is a longstanding open problem in quantum complexity theory.

quant-pharXiv:2604.11952Apr 15, 20260 citations

\mathsf{BQP}

) can be remarkably decoupled from that of classical complexity classes like

\mathsf{NP}

. Specifically: -There exists an oracle relative to which

\mathsf{NP^{BQP}}\not\subset\mathsf{BQP^{PH}}

, resolving a 2005 problem of Fortnow. As a corollary, there exists an oracle relative to which

\mathsf{P}=\mathsf{NP}

but

\mathsf{BQP}\neq\mathsf{QCMA}

. -Conversely, there exists an oracle relative to which

\mathsf{BQP^{NP}}\not\subset\mathsf{PH^{BQP}}

. -Relative to a random oracle,

\mathsf{PP}=\mathsf{PostBQP}

is not contained in the "

\mathsf{QMA}

hierarchy"

\mathsf{QMA}^{\mathsf{QMA}^{\mathsf{QMA}^{\cdots}}}

. -Relative to a random oracle,

\mathsfΣ_{k+1}^\mathsf{P}\not\subset\mathsf{BQP}^{\mathsfΣ_{k}^\mathsf{P}}

for every

k

. -There exists an oracle relative to which

\mathsf{BQP}=\mathsf{P^{\# P}}

and yet

\mathsf{PH}

is infinite. -There exists an oracle relative to which

\mathsf{P}=\mathsf{NP}\neq\mathsf{BQP}=\mathsf{P^{\# P}}

. To achieve these results, we build on the 2018 achievement by Raz and Tal of an oracle relative to which

\mathsf{BQP}\not \subset \mathsf{PH}

, and associated results about the Forrelation problem. We also introduce new tools that might be of independent interest. These include a "quantum-aware" version of the random restriction method, a concentration theorem for the block sensitivity of

\mathsf{AC^0}

circuits, and a (provable) analogue of the Aaronson-Ambainis Conjecture for sparse oracles.

Complexity class containments involving interactive proof classes are famously nonrelativizing: although

\mathsf{IP} = \mathsf{PSPACE}

, Fortnow and Sipser showed that that there exists an oracle relative to which

\mathsf{coNP} \not\subseteq \mathsf{IP}

. In contrast, the question of whether the containment

\mathsf{BQP} \subseteq \mathsf{IP}

is relativizing remains wide open. In this work we make progress towards resolving this question by showing that the containment

\mathsf{BQP} \subseteq \mathsf{MIP}

holds with respect to any classical oracle. We obtain this result by constructing, for any classical oracle

O

, a

\mathsf{PCP}

proof system for

\mathsf{BQP}^{O}

where the verifier makes polynomially many classical queries to an exponentially-long proof, and to the oracle

O

. Our construction is inspired by the state synthesis algorithm of Grover and Rudolph, and serves as a complement to the "exponential PCP" constructed by Aharonov, Arad, and Vidick, which achieves similar parameters but which is based on different ideas and does not relativize. We propose relativization as a proxy for prover efficiency, and hope that progress towards an

\mathsf{IP}

for

\mathsf{BQP}

in the oracle world will lead to a non-cryptographic interactive protocol for proving any quantum computation to a classical skeptic in the unrelativized world, which is a longstanding open problem in quantum complexity theory.