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Quantum Polynomial Hierarchies: Karp-Lipton, error reduction, and lower bounds

Avantika Agarwal, Sevag Gharibian, Venkata Koppula, Dorian Rudolph

TL;DR

This work advances the study of quantum analogues of the Polynomial-Time Hierarchy by introducing and analyzing three verifier-based quantum hierarchies: QCPH (classical proofs), QPH (mixed-state proofs), and pureQPH (pure-state proofs). It proves a collapse theorem for QCPH (QCΣ_k = QCΠ_k implies QCPH collapses to QCΣ_k) and establishes a quantum-classical Karp–Lipton theorem showing QCPH collapses to QCΣ_2 under QCMA ⊆ BQP/mpoly, with a stronger S^p_2-like analogue. For pureQPH, the authors develop one-sided error reduction using an asymmetric product test (APT) and prove that QCPH ⊆ pureQPH ⊆ EXP^PP, while also showing a nontrivial lower bound relating QCPH to QPH via purification. The results illuminate the relative power of quantum hierarchy variants and pave the way for further Toda-like containment questions, with potential implications for quantum proof systems and complexity-theoretic separations in quantum settings.

Abstract

The Polynomial-Time Hierarchy ($\mathsf{PH}$) is a staple of classical complexity theory, with applications spanning randomized computation to circuit lower bounds to ''quantum advantage'' analyses for near-term quantum computers. Quantumly, however, despite the fact that at least \emph{four} definitions of quantum $\mathsf{PH}$ exist, it has been challenging to prove analogues for these of even basic facts from $\mathsf{PH}$. This work studies three quantum-verifier based generalizations of $\mathsf{PH}$, two of which are from [Gharibian, Santha, Sikora, Sundaram, Yirka, 2022] and use classical strings ($\mathsf{QCPH}$) and quantum mixed states ($\mathsf{QPH}$) as proofs, and one of which is new to this work, utilizing quantum pure states ($\mathsf{pureQPH}$) as proofs. We first resolve several open problems from [GSSSY22], including a collapse theorem and a Karp-Lipton theorem for $\mathsf{QCPH}$. Then, for our new class $\mathsf{pureQPH}$, we show one-sided error reduction for $\mathsf{pureQPH}$, as well as the first bounds relating these quantum variants of $\mathsf{PH}$, namely $\mathsf{QCPH}\subseteq \mathsf{pureQPH} \subseteq \mathsf{EXP}^{\mathsf{PP}}$.

Quantum Polynomial Hierarchies: Karp-Lipton, error reduction, and lower bounds

TL;DR

This work advances the study of quantum analogues of the Polynomial-Time Hierarchy by introducing and analyzing three verifier-based quantum hierarchies: QCPH (classical proofs), QPH (mixed-state proofs), and pureQPH (pure-state proofs). It proves a collapse theorem for QCPH (QCΣ_k = QCΠ_k implies QCPH collapses to QCΣ_k) and establishes a quantum-classical Karp–Lipton theorem showing QCPH collapses to QCΣ_2 under QCMA ⊆ BQP/mpoly, with a stronger S^p_2-like analogue. For pureQPH, the authors develop one-sided error reduction using an asymmetric product test (APT) and prove that QCPH ⊆ pureQPH ⊆ EXP^PP, while also showing a nontrivial lower bound relating QCPH to QPH via purification. The results illuminate the relative power of quantum hierarchy variants and pave the way for further Toda-like containment questions, with potential implications for quantum proof systems and complexity-theoretic separations in quantum settings.

Abstract

The Polynomial-Time Hierarchy () is a staple of classical complexity theory, with applications spanning randomized computation to circuit lower bounds to ''quantum advantage'' analyses for near-term quantum computers. Quantumly, however, despite the fact that at least \emph{four} definitions of quantum exist, it has been challenging to prove analogues for these of even basic facts from . This work studies three quantum-verifier based generalizations of , two of which are from [Gharibian, Santha, Sikora, Sundaram, Yirka, 2022] and use classical strings () and quantum mixed states () as proofs, and one of which is new to this work, utilizing quantum pure states () as proofs. We first resolve several open problems from [GSSSY22], including a collapse theorem and a Karp-Lipton theorem for . Then, for our new class , we show one-sided error reduction for , as well as the first bounds relating these quantum variants of , namely .
Paper Structure (30 sections, 22 theorems, 38 equations, 1 figure)

This paper contains 30 sections, 22 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our results.
  3. Related work.
  4. Concurrent Work.
  5. Techniques.
  6. Discussion and open questions.
  7. Organization.
  8. Preliminaries
  9. Notation.
  10. Quantum-Classical Polynomial Hierarchy (QCPH)
  11. (Mixed-State) Quantum Polynomial Hierarchy (QPH)
  12. (Pure-State) Quantum Polynomial Hierarchy ($\mathsf{pureQPH}\xspace$)
  13. Other complexity classes
  14. Collapse theorems and Quantum Karp-Lipton
  15. Collapse Theorem for QCPH
  16. ...and 15 more sections

Key Result

theorem 1.1

If for any $k\geq 1$, $\mathsf{QC\Sigma}_k = \mathsf{QC\Pi}_k$, then $\mathsf{QCPH}\xspace = \mathsf{QC\Sigma}_k$.

Figures (1)

  • Figure 1: The SWAP test, whose output is the measurement result on the first wire.

Theorems & Definitions (52)

  • theorem 1.1
  • corollary 1.1
  • theorem 1.2
  • theorem 1.3
  • theorem 1.4
  • definition 2.1: Poly-time uniform family of quantum circuits
  • definition 2.2: $\mathsf{QC\Sigma}_i$
  • definition 2.3: $\mathsf{QC\Pi}_i$
  • definition 2.4: Quantum-Classical Polynomial Hierarchy ($\mathsf{QCPH}$)
  • definition 2.5: $\mathsf{Q\Sigma}_i$
  • ...and 42 more