Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fermionic Averaged Circuit Eigenvalue Sampling

Adrian Chapman, Steven T. Flammia

Abstract

Fermionic averaged circuit eigenvalue sampling (FACES) is a protocol to simultaneously learn the averaged error rates of many fermionic linear optical (FLO) gates simultaneously and self-consistently from a suitable collection of FLO circuits. It is highly flexible, allowing for the in situ characterization of FLO-averaged gate-dependent noise under natural assumptions on a family of continuously parameterized one- and two-qubit gates. We rigorously show that our protocol has an efficient sampling complexity, owing in-part to useful properties of the Kravchuk transformations that feature in our analysis. We support our conclusions with numerical results. As FLO circuits become universal with access to certain resource states, we expect our results to inform noise characterization and error mitigation techniques on universal quantum computing architectures which naturally admit a fermionic description.

Fermionic Averaged Circuit Eigenvalue Sampling

Abstract

Fermionic averaged circuit eigenvalue sampling (FACES) is a protocol to simultaneously learn the averaged error rates of many fermionic linear optical (FLO) gates simultaneously and self-consistently from a suitable collection of FLO circuits. It is highly flexible, allowing for the in situ characterization of FLO-averaged gate-dependent noise under natural assumptions on a family of continuously parameterized one- and two-qubit gates. We rigorously show that our protocol has an efficient sampling complexity, owing in-part to useful properties of the Kravchuk transformations that feature in our analysis. We support our conclusions with numerical results. As FLO circuits become universal with access to certain resource states, we expect our results to inform noise characterization and error mitigation techniques on universal quantum computing architectures which naturally admit a fermionic description.

Paper Structure

This paper contains 19 sections, 9 theorems, 102 equations, 2 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Fermionic Linear Optics
  4. Kravchuk Matrices
  5. Noise Channels and Twirling
  6. Channel Eigenvalues
  7. Measurement Protocol
  8. Model assumptions
  9. FLO-Specific Considerations
  10. Protocol Details
  11. Sample Complexity
  12. Numerics
  13. Conclusion
  14. Pseudocode Algorithms
  15. Proofs
  16. ...and 4 more sections

Key Result

Lemma 4

We have the alternate expressions for $M^{(\ell)}_{jk}$ Additionally, $\mathbf{M}^{(\ell)}$ is proportional to an involution

Figures (2)

  • Figure 1: A graphical outline of the FACES protocol. (a) Given a set of noisy gates $\{\mathcal{U}_i \mathcal{E}_i\}_i$ we wish to characterize, we model each such gate as the associated noise channel $\mathcal{E}_i$, followed by an application of the ideal gate $\mathcal{U}_i$. (b) We compose the gates into a collection of circuits according to the design matrix $\mathbf{A}$. (c) For each circuit $\mathcal{C}$ in the collection, we randomly sample from an ensemble of circuits whose average is $\mathcal{C}$ subject to matchgate-twirled noise. (d) We sample from the circuits in our collection to obtain an estimate for the Born-rule probabilities of the associated outcomes. (e) We apply a Kravchuk transformation to obtain the eigenvalues of each circuit. (f) We fit the gate eigenvalues to the data using a log-linear model.
  • Figure 2: Histogram of relative errors for $2n = 10$ fermionic modes with $1000$ circuits for each experiment type. See main text for description.

Theorems & Definitions (24)

  • Definition 1: Model
  • Definition 3
  • Lemma 4: Kravchuk matrix relations
  • proof
  • Definition 5
  • Lemma 6
  • proof
  • Lemma 7: FLO twirl of a Pauli channel
  • proof
  • Definition 8
  • ...and 14 more