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Modifications of Quantum Computation and Adaptive Queries to PP

David Miloschewsky, Supartha Podder

Abstract

In 2004, Aaronson introduced the complexity class $\mathsf{PostBQP}$ ($\mathsf{BQP}$ with postselection) and showed that it is equal to $\mathsf{PP}$. Following their line of work, we introduce two new complexity classes. The first, $\mathsf{CorrBQP}$, is a modification of $\mathsf{BQP}$ which has the power to perform correlated measurements, i.e. measurements that output the same value across a partition of registers. The second, $\mathsf{MajBQP}$, augments $\mathsf{BQP}$ with the ability to collapse a register to its most likely measurement outcome. Specifically, we consider two variants, $\mathsf{MajBQP}$ and $\mathsf{AdMajBQP}$, where the latter may perform intermediate measurements. We exactly characterize the computational power of the models, $\mathsf{CorrBQP} = \mathsf{AdMajBQP} = \mathsf{BPP}^{\mathsf{PP}}$ and $\mathsf{MajBQP} = \mathsf{P}^{\mathsf{PP}}$. In fact, we show that other metaphysical modifications of $\mathsf{BQP}$, such as $\mathsf{CBQP}$ (i.e. $\mathsf{BQP}$ with the ability to clone arbitrary quantum states), are also equal to $\mathsf{BPP}^{\mathsf{PP}}$. We show that $\mathsf{CorrBQP}$ and $\mathsf{MajBQP}$ are self-low with respect to classically-accessible queries. In contrast, if they were self-low under quantumly-accessible queries, the counting hierarchy would collapse. Furthermore, we introduce a variant of rational degree that lower-bounds the query complexity of $\mathsf{BPP}^{\mathsf{PP}}$. Lastly, we extend the adversary lower-bounding technique to $\mathsf{AdPDQP}$, $\mathsf{BQP}$ with the ability to sample the current state of an algorithm with collapsing it and adapt the computation based on the samples.

Modifications of Quantum Computation and Adaptive Queries to PP

Abstract

In 2004, Aaronson introduced the complexity class ( with postselection) and showed that it is equal to . Following their line of work, we introduce two new complexity classes. The first, , is a modification of which has the power to perform correlated measurements, i.e. measurements that output the same value across a partition of registers. The second, , augments with the ability to collapse a register to its most likely measurement outcome. Specifically, we consider two variants, and , where the latter may perform intermediate measurements. We exactly characterize the computational power of the models, and . In fact, we show that other metaphysical modifications of , such as (i.e. with the ability to clone arbitrary quantum states), are also equal to . We show that and are self-low with respect to classically-accessible queries. In contrast, if they were self-low under quantumly-accessible queries, the counting hierarchy would collapse. Furthermore, we introduce a variant of rational degree that lower-bounds the query complexity of . Lastly, we extend the adversary lower-bounding technique to , with the ability to sample the current state of an algorithm with collapsing it and adapt the computation based on the samples.

Paper Structure

This paper contains 32 sections, 25 theorems, 32 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Models
  3. Correlated Measurements
  4. Quantum Majority
  5. Results
  6. Related Work
  7. Preliminaries
  8. Notation
  9. Complexity Classes
  10. Complexity Measures
  11. Problems
  12. Forrelation
  13. Adaptive Construction
  14. Computational Models
  15. Correlated Measurements
  16. ...and 17 more sections

Key Result

Theorem 2.3

With respect to any classical oracle $O$,

Figures (4)

  • Figure 1: Visualization of a correlated measurement where the register $L$ is the leader and $F$ is a follower. Step (1) represents the conditioning on correlated values in the registers, while Step (2) represents the weight redistribution based on the leader register $L$.
  • Figure 2: A map of relationships between the complexity classes discussed. Given two classes $C$ and $D$, $C\rightarrow D$ indicates $C\subseteq D$. The red equal signs $\color{red}=$ signify our results. ${}^\dagger$See Result 1 for all classes equal to $\mathsf{BPP}^{\mathsf{PP}^{}\xspace}\xspace$.
  • Figure 3: Visualization of an $\mathsf{AdPDQP}^{}$ algorithm. If one removed the orange lines (i.e. the effect of $v_1$ and $v_2$), it would become a $\mathsf{PDQP}^{}$ algorithm.
  • Figure 4: Visualization of our formulation of a $\mathsf{AdPDQP}^{}$ algorithm. The orange lines represent conditioning based on the non-collapsing measurements. Note that the $C^*_i$ measurement operators are the correlated measurements where the top register is the leader. Without the orange lines, it would become a $\mathsf{PDQP}^{}$ algorithm.

Theorems & Definitions (63)

  • Definition 2.1: $\mathsf{PP}^{}$ and $\mathsf{PQP}^{}$
  • Definition 2.2: $\mathsf{PostBQP}^{}$, pp_postbqp
  • Theorem 2.3: pp_postbqp, watrous_quantum_computational_complexity
  • Definition 2.4: Counting Hierarchy
  • Definition 2.5: Quantum Circuit with Postselection and Intermediate Measurements
  • Definition 2.6: $\mathsf{AdPostBQP}^{}$ rewindable_bqp
  • Theorem 2.7: rewindable_bqp
  • Definition 2.8: $\epsilon$-almost $k$-wise Equivalency, bpppath_bqp_oracle_separation_fix
  • Theorem 2.9
  • Definition 2.10: Adaptive Construction
  • ...and 53 more