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Leibniz international proceedings in informatics

BQP After 28 Years (Invited Talk)

Scott Aaronson

Published Jan 1, 2021• 0 citations
Publisher

Abstract

I will discuss the now-ancient question of where BQP, Bounded-Error Quantum Polynomial-Time, fits in among classical complexity classes. After reviewing some basics from the 90s, I will discuss the Forrelation problem that I introduced in 2009 to yield an oracle separation between BQP and PH, and the dramatic completion of that program by Ran Raz and Avishay Tal in 2018. I will then discuss very recent work, with William Kretschmer and DeVon Ingram, which leverages the Raz-Tal theorem, along with a new "quantum-aware" random restriction method, to obtain results that illustrate just how differently BQP can behave from BPP. These include oracles relative to which NP^{BQP} ̸ ⊂ BQP^{PH} - solving a 2005 open problem of Lance Fortnow - and conversely, relative to which BQP^{NP} ̸ ⊂ PH^{BQP}; an oracle relative to which 𝖯 = NP and yet BQP ≠ QCMA; an oracle relative to which NP ⊆ BQP yet PH is infinite; an oracle relative to which 𝖯 = NP≠ BQP = PP; and an oracle relative to which PP = PostBQP ̸ ⊂ QMA^{QMA^{…}}. By popular demand, I will also speculate about the status of BQP in the unrelativized world.

DOI
10.4230/lipics.fsttcs.2021.1

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I will discuss the now-ancient question of where BQP, Bounded-Error Quantum Polynomial-Time, fits in among classical complexity classes. After reviewing some basics from the 90s, I will discuss the Forrelation problem that I introduced in 2009 to yield an oracle separation between BQP and PH, and the dramatic completion of that program by Ran Raz and Avishay Tal in 2018. I will then discuss very recent work, with William Kretschmer and DeVon Ingram, which leverages the Raz-Tal theorem, along with a new "quantum-aware" random restriction method, to obtain results that illustrate just how differently BQP can behave from BPP. These include oracles relative to which NP^{BQP} ̸ ⊂ BQP^{PH} - solving a 2005 open problem of Lance Fortnow - and conversely, relative to which BQP^{NP} ̸ ⊂ PH^{BQP}; an oracle relative to which 𝖯 = NP and yet BQP ≠ QCMA; an oracle relative to which NP ⊆ BQP yet PH is infinite; an oracle relative to which 𝖯 = NP≠ BQP = PP; and an oracle relative to which PP = PostBQP ̸ ⊂ QMA^{QMA^{…}}. By popular demand, I will also speculate about the status of BQP in the unrelativized world.
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 (BQP\mathsf{BQP}BQP) can be remarkably decoupled from that of classical complexity classes like NP\mathsf{NP}NP. Specifically: -There exists an oracle relative to which NPBQP⊄BQPPH\mathsf{NP^{BQP}}\not\subset\mathsf{BQP^{PH}}NPBQP⊂BQPPH, resolving a 2005 problem of Fortnow. As a corollary, there exists an oracle relative to which P=NP\mathsf{P}=\mathsf{NP}P=NP but BQP≠QCMA\mathsf{BQP}\neq\mathsf{QCMA}BQP=QCMA. -Conversely, there exists an oracle relative to which BQPNP⊄PHBQP\mathsf{BQP^{NP}}\not\subset\mathsf{PH^{BQP}}BQPNP⊂PHBQP. -Relative to a random oracle, PP=PostBQP\mathsf{PP}=\mathsf{PostBQP}PP=PostBQP is not contained in the "QMA\mathsf{QMA}QMA hierarchy" QMAQMAQMA⋯\mathsf{QMA}^{\mathsf{QMA}^{\mathsf{QMA}^{\cdots}}}QMAQMAQMA⋯. -Relative to a random oracle, Σk+1P⊄BQPΣkP\mathsfΣ_{k+1}^\mathsf{P}\not\subset\mathsf{BQP}^{\mathsfΣ_{k}^\mathsf{P}}Σk+1P​⊂BQPΣkP​ for every kkk. -There exists an oracle relative to which BQP=P#P\mathsf{BQP}=\mathsf{P^{\# P}}BQP=P#P and yet PH\mathsf{PH}PH is infinite. -There exists an oracle relative to which P=NP≠BQP=P#P\mathsf{P}=\mathsf{NP}\neq\mathsf{BQP}=\mathsf{P^{\# P}}P=NP=BQP=P#P. To achieve these results, we build on the 2018 achievement by Raz and Tal of an oracle relative to which BQP⊄PH\mathsf{BQP}\not \subset \mathsf{PH}BQP⊂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 AC0\mathsf{AC^0}AC0 circuits, and a (provable) analogue of the Aaronson-Ambainis Conjecture for sparse oracles.
The 222-Forrelation problem provides an optimal separation between classical and quantum query complexity and is also the problem used for separating BQP\mathsf{BQP}BQP and PH\mathsf{PH}PH relative to an oracle. A natural question is therefore to ask what are the minimal quantum resources needed to solve this problem. We show that 222-Forrelation can be solved using Instantaneous Quantum Polynomial-time (IQP\mathsf{IQP}IQP) circuits, a restricted model of quantum computation in which all gates commute. Concretely, signed 222-Forrelation can be solved by a classical random choice between two one-query IQP\mathsf{IQP}IQP circuits, while the absolute-value variant uses two independent executions of this randomized procedure. This answers a recent open question of Girish (STOC 2026) on the power of commuting quantum computations. For the Raz-Tal distribution, this randomization is unnecessary. We use this to show that there is an oracle OOO such that IQPO⊈PHO\mathsf{IQP}^O \not\subseteq \mathsf{PH}^OIQPO⊆PHO, strengthening the result of Raz and Tal (STOC 2019). It also yields an oracle separation between IQP\mathsf{IQP}IQP and DQC1\mathsf{DQC}_1DQC1​. We prove Fourier growth bounds for multi-query IQP\mathsf{IQP}IQP circuits, including bounds in terms of the size of their accepting set. Our results suggest a possible route toward decision-based quantum advantage within the restricted IQP\mathsf{IQP}IQP model. The key ingredient is an algebraic identity of the quadratic function Q(x)=∑i<jxixjQ(x) = \sum_{i < j} x_ix_jQ(x)=∑i<j​xi​xj​ that allows extracting inner-product phases within an IQP\mathsf{IQP}IQP circuit.
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 AC[superscript 0]. 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.
Alright, so now we've got this beautiful theory of quantum mechanics, and the possibly-even-more-beautiful theory of computational complexity. Clearly, with two theories this beautiful, you can't just let them stay single – you have to set them up, see if they hit it off, etc. And that brings us to the class BQP : Bounded-Error Quantum Polynomial-Time. We talked in Chapter 7 about BPP , or Bounded-Error Probabilistic Polynomial-Time. Informally, BPP is the class of computational problems that are efficiently solvable in the physical world if classical physics is true. Now we ask, what problems are efficiently solvable in the physical world if (as seems more likely) quantum physics is true? To me it’s sort of astounding that it took until the 1990s for anyone to really seriously ask this question, given that all the tools for asking it were in place by the 1960s or even earlier. It makes you wonder, what similarly obvious questions are there today that no one’s asking?
Quantum computers are widely believed have an advantage over classical computers, and some have even published some empirical evidence that this is the case. However, these publications do not include a rigorous proof of this advantage, which would have to minimally state that the class of problems decidable by a quantum computer in polynomial time, BQP, contains problems that are not in the class of problems decidable by a classical computer with similar time bounds, P. Here, I provide the proof of a stronger result that implies this result: BQP contains problems that lie beyond the much larger classical computing class NP. This proves that quantum computation is able to efficiently solve problems which are far beyond the capabilities of classical computers.