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Quantum State Compression with Polar Codes

Jack Weinberg, Avijit Mandal, Henry D. Pfister

TL;DR

This work tackles the challenge of practical quantum state compression by replacing Schumacher's typical-subspace projection with a syndrome-source coding approach based on polar codes. It introduces a quantum successive cancellation decoding framework (BPQM-inspired) to realize the encoding/decoding efficiently, and shows that the scheme achieves asymptotic rate-optimality with vanishing failure probability while approaching the Schumacher limit $S(\rho)$. The authors provide a concrete protocol, a lifted BP-based decoding strategy, and a length-4 toy example, along with numerical results at short blocklengths that demonstrate clear advantages over Schumacher compression. The approach promises scalable, low-complexity quantum data compression with provable rate guarantees, potentially benefiting quantum storage and communication systems.

Abstract

In the quantum compression scheme proposed by Schumacher, Alice compresses a message that Bob decompresses. In that approach, there is some probability of failure and, even when successful, some distortion of the state. For sufficiently large blocklengths, both of these imperfections can be made arbitrarily small while achieving a compression rate that asymptotically approaches the source coding bound. However, direct implementation of Schumacher compression suffers from poor circuit complexity. In this paper, we consider a slightly different approach based on classical syndrome source coding. The idea is to use a linear error-correcting code and treat the message to be compressed as an error pattern. If the message is a correctable error (i.e., a coset leader) then Alice can use the error-correcting code to convert her message to a corresponding quantum syndrome. An implementation of this based on polar codes is described and simulated. As in classical source coding based on polar codes, Alice maps the information into the ``frozen" qubits that constitute the syndrome. To decompress, Bob utilizes a quantum version of successive cancellation coding.

Quantum State Compression with Polar Codes

TL;DR

This work tackles the challenge of practical quantum state compression by replacing Schumacher's typical-subspace projection with a syndrome-source coding approach based on polar codes. It introduces a quantum successive cancellation decoding framework (BPQM-inspired) to realize the encoding/decoding efficiently, and shows that the scheme achieves asymptotic rate-optimality with vanishing failure probability while approaching the Schumacher limit . The authors provide a concrete protocol, a lifted BP-based decoding strategy, and a length-4 toy example, along with numerical results at short blocklengths that demonstrate clear advantages over Schumacher compression. The approach promises scalable, low-complexity quantum data compression with provable rate guarantees, potentially benefiting quantum storage and communication systems.

Abstract

In the quantum compression scheme proposed by Schumacher, Alice compresses a message that Bob decompresses. In that approach, there is some probability of failure and, even when successful, some distortion of the state. For sufficiently large blocklengths, both of these imperfections can be made arbitrarily small while achieving a compression rate that asymptotically approaches the source coding bound. However, direct implementation of Schumacher compression suffers from poor circuit complexity. In this paper, we consider a slightly different approach based on classical syndrome source coding. The idea is to use a linear error-correcting code and treat the message to be compressed as an error pattern. If the message is a correctable error (i.e., a coset leader) then Alice can use the error-correcting code to convert her message to a corresponding quantum syndrome. An implementation of this based on polar codes is described and simulated. As in classical source coding based on polar codes, Alice maps the information into the ``frozen" qubits that constitute the syndrome. To decompress, Bob utilizes a quantum version of successive cancellation coding.
Paper Structure (13 sections, 2 theorems, 25 equations, 2 figures)

This paper contains 13 sections, 2 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Binary Linear Codes
  4. Polar Codes
  5. Syndrome Source Coding
  6. Quantum Formalism
  7. Quantum Compression by Schumacher
  8. Quantum Compression via Syndrome Source Coding
  9. Protocol
  10. Efficient Implementation
  11. Decoding a length-4 polar code
  12. Main Results
  13. Numerical Results

Key Result

Proposition 1

The proposed protocol is successful if and only if Alice measures the outcome associated with projector $\Pi^N_K$, which occurs with probability $\Tr(\Pi^N_K \rho^{\otimes N})$.

Figures (2)

  • Figure 1: Quantum compression with embedded polar codes for blocklength 4.
  • Figure :

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof