bibkey: sudmarshallwangrieffelwudarski2022dual
authors: James Sud; Jeffrey Marshall; Zhihui Wang; Eleanor Rieffel; Filip A. Wudarski
year: 2022
title: "Dual-map framework for noise characterization of quantum computers"
doi: 10.1103/PhysRevA.106.012606
url: https://arxiv.org/abs/2112.04414v1
claim: "The paper introduces a dual-map framework for noisy expectation values, defines the marginal approximation to the effective noise (MATEN) as the tensor product of the single-qubit marginals of the noise channel, evaluates it on QAOA circuits and on Rigetti hardware, computes that the maximally entangled process matrix gives fidelity 4^-N with its marginal approximation, and conjectures that this process matrix is the worst case for the MATEN protocol."
strata_touched:
  - D5/S3/Quantum/QuantumChannels/MarginalApproximationWorstFidelity
license: citation-only
triage: anchor

Dual-map framework for noise characterization of quantum computers

J. Sud, J. Marshall, Z. Wang, E. Rieffel and F. A. Wudarski, arXiv:2112.04414v1; Phys. Rev. A 106, 012606 (2022). Quotations are from the v1 TeX source.

The marginal approximation:

we define a marginal approximation to the effective noise (MATEN) as $\widetilde{\text{ep}}={⨂}_{k=1}^N{\mathrm{tr}}_{\bar{k}}\left(\text{ep}\right)={⨂}_{k=1}^N{\text{ep}}_k,$ where ${\mathrm{tr}}_{\bar{k}}(\text{ep})={\text{ep}}_k$ traces out all subsystem except $k$-th

The process matrix:

${\mathcal{E}}_{\chi }(\rho )={\sum }_{k,l=0}^3{\chi }_{kl}{\sigma }_k\rho {\sigma }_l,$ where ${\chi }_{kl}$ are elements of the $\chi$ matrix

The maximally entangled process matrix and the conjecture:

$\chi =|\Psi ⟩⟨\Psi |\to {\chi }_{\mathrm{MA}}={⨂}_{k=1}^N\left(\frac{1}{4}\text{mathbbm}1\right)=\frac{1}{4^N}\text{mathbbm}{1}_{4^N},$ […] Above we denote the non-local process matrix $\chi$ as the maximally entangled state, that is defined as a projector onto $|\Psi ⟩=\frac{1}{2}{\sum }_{i=0}^3|ii\dots i⟩$, with $N$ 4D subsystems (each corresponding to a qubit), note that this is a GHZ state \cite{greenberger2007going}. We conjecture that the effective channel with the maximally entangled $\chi$ matrix is the worst case scenario for the proposed MATEN protocol.

However, this conjecture requires more rigorous treatment, which we leave as an open problem. Note that maximally entangled $\chi$ is a completely valid choice, since $\chi \ge 0$ and the map associated with it is trace preserving.

The fidelity:

$F\left(\chi ,{\chi }_{MA}\right)=\frac{1}{4^N}\mathrm{tr}(\sqrt{\chi }{)}^2=\frac{1}{4^N},$

Verified locator

Source: https://arxiv.org/abs/2112.04414v1

• URL: https://arxiv.org/abs/2112.04414v1 (source main.tex, md5 b2275e7db7e3a63cc4c871c172f7943a):
  • the marginal approximation, l. 226–232;
  • the process matrix, l. 234–237;
  • the maximally entangled process matrix and the conjecture, l. 284–290;
  • the fidelity, l. 292–296.
• DOI: 10.1103/PhysRevA.106.012606 (https://doi.org/10.1103/PhysRevA.106.012606).