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The Space-Time Cost of Purifying Quantum Computations

Mark Zhandry

TL;DR

This paper addresses whether intermediate quantum measurements can be eliminated from quantum computations in a way that is simultaneously space- and time-efficient. It introduces a black-box purifier framework and proves a fundamental barrier: any such purifier must incur either a space blow-up proportional to the running time or an exponential time blow-up in the space budget, under a black-box model. The authors develop a stateful-simulation technique and an oracle-based separation to show that, for low-space computations with measurements, there exist unitary simulations that necessarily require space Ω(T) or time 2^{Ω(S)}, ruling out efficient black-box purification across regimes. The results clarify why previous space-efficient transformations incur time blowups and establish a robust barrier against circuit-level purification approaches that do not exploit specific structure of the computation. Overall, the work formalizes the limitations of purifying quantum computations and highlights the intrinsic tradeoffs between space and time in the presence of measurements.

Abstract

General quantum computation consists of unitary operations and also measurements. It is well known that intermediate quantum measurements can be deferred to the end of the computation, resulting in an equivalent purely unitary computation. While time efficient, this transformation blows up the space to linear in the running time, which could be super-polynomial for low-space algorithms. Fefferman and Remscrim (STOC'21) and Girish, Raz and Zhan (ICALP'21) show different transformations which are space efficient, but blow up the running time by a factor that is exponential in the space. This leaves the case of algorithms with small-but-super-logarithmic space as incurring a large blowup in either time or space complexity. We show that such a blowup is likely inherent, demonstrating that any "black-box" transformation which removes intermediate measurements must significantly blow up either space or time.

The Space-Time Cost of Purifying Quantum Computations

TL;DR

This paper addresses whether intermediate quantum measurements can be eliminated from quantum computations in a way that is simultaneously space- and time-efficient. It introduces a black-box purifier framework and proves a fundamental barrier: any such purifier must incur either a space blow-up proportional to the running time or an exponential time blow-up in the space budget, under a black-box model. The authors develop a stateful-simulation technique and an oracle-based separation to show that, for low-space computations with measurements, there exist unitary simulations that necessarily require space Ω(T) or time 2^{Ω(S)}, ruling out efficient black-box purification across regimes. The results clarify why previous space-efficient transformations incur time blowups and establish a robust barrier against circuit-level purification approaches that do not exploit specific structure of the computation. Overall, the work formalizes the limitations of purifying quantum computations and highlights the intrinsic tradeoffs between space and time in the presence of measurements.

Abstract

General quantum computation consists of unitary operations and also measurements. It is well known that intermediate quantum measurements can be deferred to the end of the computation, resulting in an equivalent purely unitary computation. While time efficient, this transformation blows up the space to linear in the running time, which could be super-polynomial for low-space algorithms. Fefferman and Remscrim (STOC'21) and Girish, Raz and Zhan (ICALP'21) show different transformations which are space efficient, but blow up the running time by a factor that is exponential in the space. This leaves the case of algorithms with small-but-super-logarithmic space as incurring a large blowup in either time or space complexity. We show that such a blowup is likely inherent, demonstrating that any "black-box" transformation which removes intermediate measurements must significantly blow up either space or time.
Paper Structure (43 sections, 22 theorems, 25 equations, 5 figures)

This paper contains 43 sections, 22 theorems, 25 equations, 5 figures.

Table of Contents

  1. Introduction
  2. What is a Quantum Measurement, Anyway?
  3. Relationship to Classical Reversible Computation
  4. Our Results
  5. Formalizing Black Box Impossibilities.
  6. Our Main Theorem.
  7. New Space Lower Bound Technique.
  8. Technical Overview
  9. Our Construction.
  10. Formalizing Black Box Compilers.
  11. Proving Large Unitary Space.
  12. Simulating ${\mathbf{O}}$ Statefully.
  13. Preliminaries
  14. Quantum Computation.
  15. Distance.
  16. ...and 28 more sections

Key Result

Theorem 1.1

For any black-box compiler mapping space $S$, time $T$ general quantum computation to space $S'$, time $T'$ unitary computation, either $S'=\Omega(T)$ or $T'=2^{\Omega(S)}$.

Figures (5)

  • Figure 1: Different kinds of measurement gates. Here, $|\psi\rangle$ is the state being measured, $b$ is the probabilistic measurement outcome, and $|\phi\rangle$ is the state that $|\psi\rangle$ collapses to when the measurement outcome is $b$.
  • Figure 2: Our task that can be computed in low time and space with measurements, but requires large space or time without.
  • Figure 3: Our task that can be computed in low space and time with measurements, but requires large space or large time without.
  • Figure 4: Converting from an algorithm ${\mathcal{A}}$ (top) which computes ${\sf out}$ to an algorithm ${\mathcal{A}}_1$ (bottom) which computes $|\psi_t\rangle$.
  • Figure 5: How our simulator maps the query input (top) to the query output (bottom) by simply moving registers around.

Theorems & Definitions (49)

  • Remark 1
  • Theorem 1.1: Informal
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Lemma 3.1: BBBV97 Theorem 3.1
  • Lemma 3.2: BBBV97 Theorem 3.3
  • Definition 4.1
  • Remark 6
  • ...and 39 more