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An access model for quantum encoded data

Miguel Murça, Paul K. Faehrmann, Yasser Omar

TL;DR

This work introduces and investigates a data access model that is satisfiable by the preparation and measurement of block encoded states, as well as in contexts such as classical quantum circuit simulation or Pauli sampling, and illustrates that this abstraction is compositional and has some computational power.

Abstract

We introduce and investigate a data access model (approximate sample and query) that is satisfiable by the preparation and measurement of block encoded states, as well as in contexts such as classical quantum circuit simulation or Pauli sampling. We illustrate that this abstraction is compositional and has some computational power. We then apply these results to obtain polynomial improvements over the state of the art in the sample and computational complexity of distributed inner product estimation. By doing so, we provide a new interpretation for why Pauli sampling is useful for this task. Our results partially characterize the power of time-limited fault-tolerant quantum circuits aided by classical computation. They are a first step towards extending the classical data Quantum Singular Value Transform dequantization results to a quantum setting.

An access model for quantum encoded data

TL;DR

This work introduces and investigates a data access model that is satisfiable by the preparation and measurement of block encoded states, as well as in contexts such as classical quantum circuit simulation or Pauli sampling, and illustrates that this abstraction is compositional and has some computational power.

Abstract

We introduce and investigate a data access model (approximate sample and query) that is satisfiable by the preparation and measurement of block encoded states, as well as in contexts such as classical quantum circuit simulation or Pauli sampling. We illustrate that this abstraction is compositional and has some computational power. We then apply these results to obtain polynomial improvements over the state of the art in the sample and computational complexity of distributed inner product estimation. By doing so, we provide a new interpretation for why Pauli sampling is useful for this task. Our results partially characterize the power of time-limited fault-tolerant quantum circuits aided by classical computation. They are a first step towards extending the classical data Quantum Singular Value Transform dequantization results to a quantum setting.

Paper Structure

This paper contains 20 sections, 18 theorems, 79 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Prior art
  3. Our contributions
  4. Notation and preliminaries
  5. Notation
  6. Block encodings and projective encodings
  7. One-sided and two-sided error
  8. Sample and Query
  9. Singly coherent copy tomography
  10. Simulation of low T-gate count states
  11. Pauli sampling and nonstabilizerness
  12. Approximate Sample and Query
  13. Satisfying approximate sample and query access
  14. Composing approximate oversample and query access
  15. Computation with approximate oversample and query access
  16. ...and 5 more sections

Key Result

Lemma 1.1

(Linear combinations of ASQ --- Informal statement of Lemmas lemma:lin-comb and lemma:lin-comb-rand.) Given $\text{ASQ}_{\phi}$ access to complex vectors $x_1, \ldots, x_\tau$, and complex coefficients $\lambda_1, \ldots, \lambda_\tau$, one may obtain $\text{ASQ}_{\phi'}$ access (with certainty) to

Figures (4)

  • Figure 1: The approximate sample and query (ASQ) access model restricts access to a vector to three rules (Fig. \ref{['subfig:asq']}). These can be satisfied by quantum or classical agents in the right conditions. By developing algorithms that operate over this abstraction, their runtime can be described in terms of the time necessary to carry out each access operation. If a classical agent can satisfy each operation fast enough, a dequantization result follows. In this paper, we establish some of the power of ASQ (Fig. \ref{['subfig:power']}): we show that an agent can compose ASQ into linear combinations, and that they can use ASQ access to compute the inner product.
  • Figure : Calculating a relative error estimate via online access to randomized queries to absolute error estimates.
  • Figure : Calculating an inner product estimate via $\text{ASQ}_{\phi}$ access to one of the vectors ($x$), and classical RAM access to the other ($y$), with success probability at least $2/3$, and absolute error at most $\epsilon\lVert y\rVert_1$.
  • Figure : Calculating an inner product estimate via $\text{ASQ}_{\phi}$ access to both vectors ($x$ and $y$), with success probability at least $2/3$ and absolute error at most $\epsilon [1 + \min(\frac{\lVert x\rVert_1}{\lVert x\rVert}, \frac{\lVert y\rVert_1}{\lVert y\rVert})]$.

Theorems & Definitions (48)

  • Definition 1
  • definition 1
  • definition 2
  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • definition 3
  • definition 4
  • definition 5
  • definition 6
  • ...and 38 more