Circuit Extraction for ZX-diagrams can be #P-hard

• URL: http://arxiv.org/abs/2202.09194v1
• Date: Fri, 18 Feb 2022 13:50:24 GMT
• Title: Circuit Extraction for ZX-diagrams can be #P-hard
• Authors: Niel de Beaudrap, Aleks Kissinger, John van de Wetering
• Abstract summary: Some applications for the ZX-calculus rely on being able to efficiently translate a ZX-diagram back into a quantum circuit of comparable size. In this paper we show that the circuit extraction problem is #P-hard, and so is itself at least as hard as strong simulation of quantum circuits.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The ZX-calculus is a graphical language for reasoning about quantum computation using ZX-diagrams, a certain flexible generalisation of quantum circuits that can be used to represent linear maps from $m$ to $n$ qubits for any $m,n \geq 0$. Some applications for the ZX-calculus, such as quantum circuit optimisation and synthesis, rely on being able to efficiently translate a ZX-diagram back into a quantum circuit of comparable size. While several sufficient conditions are known for describing families of ZX-diagrams that can be efficiently transformed back into circuits, it has previously been conjectured that the general problem of circuit extraction is hard. That is, that it should not be possible to efficiently convert an arbitrary ZX-diagram describing a unitary linear map into an equivalent quantum circuit. In this paper we prove this conjecture by showing that the circuit extraction problem is #P-hard, and so is itself at least as hard as strong simulation of quantum circuits. In addition to our main hardness result, which relies specifically on the circuit representation, we give a representation-agnostic hardness result. Namely, we show that any oracle that takes as input a ZX-diagram description of a unitary and produces samples of the output of the associated quantum computation enables efficient probabilistic solutions to NP-complete problems.

Related papers

• The Basis of Design Tools for Quantum Computing: Arrays, Decision Diagrams, Tensor Networks, and ZX-Calculus [55.58528469973086]
  Quantum computers promise to efficiently solve important problems classical computers never will. A fully automated quantum software stack needs to be developed. This work provides a look "under the hood" of today's tools and showcases how these means are utilized in them, e.g., for simulation, compilation, and verification of quantum circuits.
  arXiv  Detail & Related papers  (2023-01-10T19:00:00Z)
• Compilation of algorithm-specific graph states for quantum circuits [55.90903601048249]
  We present a quantum circuit compiler that prepares an algorithm-specific graph state from quantum circuits described in high level languages. The computation can then be implemented using a series of non-Pauli measurements on this graph state.
  arXiv  Detail & Related papers  (2022-09-15T14:52:31Z)
• Vanishing 2-Qubit Gates with Non-Simplification ZX-Rules [1.0089382889894247]
  A quantum circuit can be translated to a ZX-diagram which can be simplified using the rules of the ZX-calculus. The best-known extraction procedures can drastically increase the number of 2-qubit gates. We take advantage of the fact that local changes in a ZX-diagram can drastically affect the complexity of the extracted circuit.
  arXiv  Detail & Related papers  (2022-09-14T18:43:21Z)
• Equivalence Checking of Quantum Circuits with the ZX-Calculus [3.610459670994051]
  State-of-the-art quantum computers are capable of running increasingly complex algorithms. The need for automated methods to design and test potential applications rises. Recently, new methods have been proposed that tackle this problem.
  arXiv  Detail & Related papers  (2022-08-26T18:00:01Z)
• Diagrammatic Analysis for Parameterized Quantum Circuits [0.0]
  We describe extensions of the ZX-calculus especially suitable for parameterized quantum circuits. We provide several new ZX-diagram rewrite rules and generalizations for this setting. We demonstrate that the diagrammatic approach offers useful insights into algorithm structure and performance.
  arXiv  Detail & Related papers  (2022-04-04T08:26:20Z)
• Resource Optimisation of Coherently Controlled Quantum Computations with the PBS-calculus [55.2480439325792]
  Coherent control of quantum computations can be used to improve some quantum protocols and algorithms. We refine the PBS-calculus, a graphical language for coherent control inspired by quantum optics.
  arXiv  Detail & Related papers  (2022-02-10T18:59:52Z)
• An Algebraic Quantum Circuit Compression Algorithm for Hamiltonian Simulation [55.41644538483948]
  Current generation noisy intermediate-scale quantum (NISQ) computers are severely limited in chip size and error rates. We derive localized circuit transformations to efficiently compress quantum circuits for simulation of certain spin Hamiltonians known as free fermions. The proposed numerical circuit compression algorithm behaves backward stable and scales cubically in the number of spins enabling circuit synthesis beyond $mathcalO(103)$ spins.
  arXiv  Detail & Related papers  (2021-08-06T19:38:03Z)
• Simplification Strategies for the Qutrit ZX-Calculus [0.0]
  The ZX-calculus is a graphical language for suitably represented tensor networks, called ZX-diagrams. The ZX-calculus has found applications in reasoning about quantum circuits, condensed matter systems, quantum algorithms, quantum error codes, and counting problems.
  arXiv  Detail & Related papers  (2021-03-11T19:17:28Z)
• Quantum Garbled Circuits [9.937090317971313]
  We show how to compute an encoding of a given quantum circuit and quantum input, from which it is possible to derive the output of the computation and nothing else. Our protocol has the so-called $Sigma$ format with a single-bit challenge, and allows the inputs to be delayed to the last round.
  arXiv  Detail & Related papers  (2020-06-01T17:07:01Z)
• On estimating the entropy of shallow circuit outputs [49.1574468325115]
  Estimating the entropy of probability distributions and quantum states is a fundamental task in information processing. We show that entropy estimation for distributions or states produced by either log-depth circuits or constant-depth circuits with gates of bounded fan-in and unbounded fan-out is at least as hard as the Learning with Errors problem.
  arXiv  Detail & Related papers  (2020-02-27T15:32:08Z)
• PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations [77.34726150561087]
  We introduce the PBS-calculus to represent and reason on quantum computations involving coherent control of quantum operations. We equip the language with an equational theory, which is proved to be sound and complete. We consider applications like the implementation of controlled permutations and the unrolling of loops.
  arXiv  Detail & Related papers  (2020-02-21T16:15:58Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.