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We study the problem of finding a (pure) product state with optimal fidelity to an unknown n-qubit quantum state ρ, given copies of ρ. This is a basic instance of a fundamental question in quantum learning: is it possible to efficiently learn a simple approximation to an arbitrary state? We give an algorithm which finds a product state with fidelity ε-close to optimal, using N=npoly(1/ε) copies of ρ and poly(N) classical overhead. We further show that estimating the optimal fidelity is NP-hard for error ε=1/poly(n), showing that the error dependence cannot be significantly improved. For our algorithm, we build a carefully-defined cover over candidate product states, qubit by qubit, and then demonstrate that extending the cover can be reduced to approximate constrained polynomial optimization. For our proof of hardness, we give a formal reduction from polynomial optimization to finding the closest product state. Together, these results demonstrate a fundamental connection between these two seemingly unrelated questions. Building on our general approach, we also develop more efficient algorithms in three simpler settings: when the optimal fidelity exceeds 5/6; when we restrict ourselves to a discrete class of product states; and when we are allowed to output a matrix product state.
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On the Hardness of Approximating Distances of Quantum Codes
The problem of computing distances of error-correcting codes is fundamental in both the classical and quantum settings. While hardness for the classical version of these problems has been known for some time (in both the exact and approximate settings), it was only recently that Kapshikar and Kundu showed these problems are also hard in the quantum setting. As our first main result, we reprove this using arguably simpler arguments based on hypergraph product codes. In particular, we get a direct reduction to CSS codes, the most commonly used type of quantum code, from the minimum distance problem for classical linear codes. Our second set of results considers the distance of a graph state, which is a key parameter for quantum codes obtained via the codeword stabilized formalism. We show that it is NP-hard to compute/approximate the distance of a graph state when the adjacency matrix of the graph is the input. In fact, we show this is true even if we only consider X-type errors of a graph state. Our techniques moreover imply an interesting classical consequence: the hardness of computing or approximating the distance of classical codes with rate equal to 1/2. One of the main motivations of the present work is a question raised by Kapshikar and Kundu concerning the NP-hardness of approximation when there is an additive error proportional to a quantum code’s length. We show that no such hardness can hold for hypergraph product codes. These observations suggest the possibility of a new kind of square root barrier.
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- Award ID(s):
- 2330130
- PAR ID:
- 10658279
- Editor(s):
- Aiswarya, C; Mehta, Ruta; Roy, Subhajit
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 360
- ISSN:
- 1868-8969
- Page Range / eLocation ID:
- 34:1-34:18
- Subject(s) / Keyword(s):
- quantum codes minimum distance problem NP-hardness graph state distance Theory of computation → Error-correcting codes
- Format(s):
- Medium: X Size: 18 pages; 802802 bytes Other: application/pdf
- Size(s):
- 18 pages 802802 bytes
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/lipics.fsttcs.2025.34
