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New Lower-bounds for Quantum Computation with Non-Collapsing Measurements

David Miloschewsky, Supartha Podder

TL;DR

This paper analyzes the power of non-collapsing measurements in quantum computation by studying the PDQP model. It introduces an equivalent circuit-based definition to enable rigorous lower-bounding using a positive weighted adversary method and derives a quantitative trade-off between the number of queries $Q$ and non-collapsing measurements $P$, yielding bounds such as $QP = \,\Omega\(\sqrt{mm'/ll'}\)$ and, in the non-adaptive setting, $QP = \,\Omega\(\max\{m/l, m'/l'\}\)$. Applying these bounds to unstructured search, majority, parity, collision, and element distinctness, the authors obtain tight or near-tight results, including a tight $\Theta\(N^{1/3}\)\) bound for search in PDQP and $\Theta\(\sqrt{N}\)$ in the non-adaptive variant, while showing constant-time results for collision and size-bounded behavior for element distinctness. They further explore PDQP under query restrictions and compare with copy-enabled models (CBQP), highlighting the distinct advantages provided by non-collapsing measurements versus copying. The work advances a framework for understanding quantum speedups under non-standard measurement capabilities and identifies several open questions, including the adaptation of lower-bound techniques and polynomial-method approaches to PDQP. Overall, the paper clarifies the limited speedups PDQP offers over BQP and maps the landscape of lower bounds across multiple problem families in this metaphysical quantum setting.

Abstract

Aaronson, Bouland, Fitzsimons and Lee introduced the complexity class PDQP (which was original labeled naCQP), an alteration of BQP enhanced with the ability to obtain non-collapsing measurements, samples of quantum states without collapsing them. Although PDQP contains SZK, it still requires $Ω(N^{1/4})$ queries to solve unstructured search. We formulate an alternative equivalent definition of PDQP, which we use to prove the positive weighted adversary lower-bounding method, establishing multiple tighter bounds and a trade-off between queries and non-collapsing measurements. We utilize the technique in order to analyze the query complexity of the well-studied majority and element distinctness problems. Additionally, we prove a tight $Θ(N^{1/3})$ bound on search. Furthermore, we use the lower-bound to explore PDQP under query restrictions, finding that when combined with non-adaptive queries, we limit the speed-up in several cases.

New Lower-bounds for Quantum Computation with Non-Collapsing Measurements

TL;DR

This paper analyzes the power of non-collapsing measurements in quantum computation by studying the PDQP model. It introduces an equivalent circuit-based definition to enable rigorous lower-bounding using a positive weighted adversary method and derives a quantitative trade-off between the number of queries and non-collapsing measurements , yielding bounds such as and, in the non-adaptive setting, . Applying these bounds to unstructured search, majority, parity, collision, and element distinctness, the authors obtain tight or near-tight results, including a tight \Theta$ in the non-adaptive variant, while showing constant-time results for collision and size-bounded behavior for element distinctness. They further explore PDQP under query restrictions and compare with copy-enabled models (CBQP), highlighting the distinct advantages provided by non-collapsing measurements versus copying. The work advances a framework for understanding quantum speedups under non-standard measurement capabilities and identifies several open questions, including the adaptation of lower-bound techniques and polynomial-method approaches to PDQP. Overall, the paper clarifies the limited speedups PDQP offers over BQP and maps the landscape of lower bounds across multiple problem families in this metaphysical quantum setting.

Abstract

Aaronson, Bouland, Fitzsimons and Lee introduced the complexity class PDQP (which was original labeled naCQP), an alteration of BQP enhanced with the ability to obtain non-collapsing measurements, samples of quantum states without collapsing them. Although PDQP contains SZK, it still requires queries to solve unstructured search. We formulate an alternative equivalent definition of PDQP, which we use to prove the positive weighted adversary lower-bounding method, establishing multiple tighter bounds and a trade-off between queries and non-collapsing measurements. We utilize the technique in order to analyze the query complexity of the well-studied majority and element distinctness problems. Additionally, we prove a tight bound on search. Furthermore, we use the lower-bound to explore PDQP under query restrictions, finding that when combined with non-adaptive queries, we limit the speed-up in several cases.

Paper Structure

This paper contains 25 sections, 16 theorems, 50 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Results
  4. Lower-bounding methods
  5. Applications of the lower-bounding methods
  6. Proof techniques
  7. Alternative definition of computation with non-collapsing measurements
  8. Obtaining the lower-bounding techniques
  9. Related work
  10. Open problems
  11. Preliminaries
  12. Quantum Query Model
  13. Non-collapsing measurement model
  14. Lower-bounds on non-collapsing measurements
  15. Weighted adversary method
  16. ...and 10 more sections

Key Result

Lemma 2.1

Let $\ket{\psi_t}$ and $\ket{\psi_t^x}$ be states of an algorithm solving search using $Q$ queries in total after $t$ steps without partial measurement where respectively there is no marked element or $x\in [N]$ is the marked element. Then,

Figures (4)

  • Figure 1: Example of the process of the oracle $\mathcal{Q}_P$ with $P=3$. Given a circuit $C$, the oracle $\mathcal{Q}_P$ returns $\{v_i\}_i^3$ by measuring the states $\{\phi_i\}_i^3$ at each step of $C$.
  • Figure 2: Example of the circuit $C^*$ with $P=3$ which is run by oracle $\mathcal{Q}_P^*$, producing the same results as oracle $\mathcal{Q}_P$.
  • Figure 3: Examples of the quantum oracle model with non-collapsing measurements based on (a) Definition \ref{['definition_pdqp_original']} and (b) Definition \ref{['definition_pdqp_our']}.
  • Figure 4: Example of purifying a $\mathsf{CBQP}^{}$ circuit.

Theorems & Definitions (32)

  • Lemma 2.1: Hybrid argument of bbbv_97
  • Definition 2.2: Weight scheme of weighted_adversary
  • Definition 2.3: Weight load of weighted_adversary
  • Proposition 2.4: Weight identity, proof of Lemma 4 of weighted_adversary
  • Definition 3.1: Section 2 of space_above_bqp
  • Proposition 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • proof
  • ...and 22 more