New technique: Tensor-network error mitigation

[Misty-W]

Issue Description

Inspired by the Tensor-network error mitigation (TEM) technique in Scalable tensor-network error mitigation for near-term quantum computing by S. Filippov, M. Leahy, M. Rossi, G. García-Pérez (arXiv:23077.11740).
TEM is a post-processing error mitigation technique, similar to PEC in the construction and application of an inverted noise channel, but dissimilar in that it does not require sampling many noisy circuit instances.

Proposed Solution

TEM consists of two main parts:

1. Construct a tensor network representing the inverse of the global noise channel affecting the state of the quantum processor
2. Apply the map to measurement outcomes obtained from the noisy state.

Considerations:
According to the results reported in https://arxiv.org/abs/2307.11740,

1. TEM does not require additional quantum operations other than the implementation of informationally complete POVMs, obtained through randomized local measurements --> TEM measurement overhead is quadratically smaller than in PEC.
2. TEM can be applied to circuits of twice the depth compared to what is achievable with PEC under realistic conditions with sparse Pauli-Lindblad noise, such as those in [E. van den Berg et al., Nat. Phys. (2023)]

Additional References

[Misty-W]

Assigned myself to figure out difficulty of implementation and possible impact to dependencies.

[natestemen]

Paper on scalability of QEM techniques, including TEM: https://arxiv.org/abs/2403.13542

[nathanshammah]

@Misty-W this is a very interesting technique, also contains quite complex elements. I propose we schedule Mitiq discussions/Quantum Wednesday talk on it to review the theory, results and methods, and then based on findings decide on next steps, e.g., RFC. Discussed this with @FarLab who can help prioritize in the next week.

[FarLab]

@nathanshammah @natestemen @Misty-W @jordandsullivan @cosenal
We could try and understand the details of the paper and how to potentially implement it by discussing/brainstorming during the Tuesday coding sessions. I propose we all just take a look at the paper before that and discuss things we understand or don't (you don't have to understand every detail by Tuesday!). Let me know what you think.

[jordandsullivan]

Sounds great to me.

…

On Thu, Apr 11, 2024 at 10:32 AM FarLab ***@***.***> wrote: @nathanshammah <https://github.com/nathanshammah> @natestemen <https://github.com/natestemen> @Misty-W <https://github.com/Misty-W> @jordandsullivan <https://github.com/jordandsullivan> @cosenal <https://github.com/cosenal> We could try and understand the details of the paper and how to potentially implement it by discussing/brainstorming during the Tuesday coding sessions. I propose we all just take a look at the paper before that and discuss things we understand or don't (you don't have to understand every detail by Tuesday!). Let me know what you think. — Reply to this email directly, view it on GitHub <#2197 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/ADYYB52F5ZFSA3HFILT2CBTY43CJFAVCNFSM6AAAAABDTZBDVKVHI2DSMVQWIX3LMV43OSLTON2WKQ3PNVWWK3TUHMZDANJQGE3TIMJWHA> . You are receiving this because you were mentioned.Message ID: ***@***.***>

-- Best, Jordan

[FarLab]

A small summary of TEM:

1. Given the circuit of interest, use a IC-POVM to perform tomography and obtain an estimate of the noisy density matrix $\tilde{\rho}$.
2. If the noise is reasonably small and the inverse of the noise channel has an compact tensor network representation (small bond dimension), we could run the inverse of the noisy circuit applied to $\tilde{\rho}$ on a classical computer followed by running the ideal circuit. This part is done efficiently by contracting the tensor network from the "middle out". The result is an estimate of the ideal density matrix $\hat{\rho}$.
3. Compute the expectation value of the observable $Tr(\hat{\rho} O )$ (also done using tensor networks).

The effectiveness of this error mitigation scheme hinges entirely on step 2), since simulating a (noisy) circuit using tensor networks is exponentially hard in general. But in the paper they show that in the case that noise is sufficiently small (so the noise channel is invertible) and its inverse has an efficient tensor network representation, then the contraction of the inverse of the noisy circuit followed immediately by the noiseless circuit can be done efficiently.

It looks like Algorithmiq has patented the TEM algorithm (correct me I am wrong @nathanshammah), so we have to figure out first if we can/are allowed to implement this algorithm in Mitiq. So for this milestone we will close this issue, but possible to reopen again once we know more about this.

[purva-thakre]

Found a paper that relies on tensor networks for noise characterization. The authors also combine their noise characterization technique with tensor network error mitigation as the characterization output is particularly suited for TEM.

https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.6.033217

Our method not only avoids the exponential
scaling of measurement resources by sampling the different
possible tomographic settings in a randomized way, but it
also enables an efficient, meaningful, and scalable descrip-
tion of the reconstructed noise channel by means of tensor
network techniques (more specifically, a locally purified den-
sity operator structure, LPDO) with low bond dimension.