Mitiq PEC Planning

August 2020

General PEC Pipeline

Input:

(i) Circuit C

(ii) Executor E

(iii) Number of samples M or precision δ

(1) Compile circuit C into a set of unitaries {U}

(2) Express each unitary in {U} in the basis of noisy gates {G}

Compute or input: Noisy basis {G}

Output:

Noiseless estimate E(C)

(i) Sample a noisy circuit N from the quasi-probability distribution

(ii) Use the executor to evaluate E(N) and store the result + sign and prefactor

Breakdown of steps

• PEC requires each gate in the circuit C to be written in a (noisy) basis {G}.
• It is easiest to express every gate in the basis if C has been compiled into a (small) set of gates {U}.
• In this case a simple dictionary mapping can be used.

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Note: If we assume a particular noise model, the noisy basis {G} can have a simple representation for any unitary in the circuit C, so compilation may be overkill here.

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(1) Compile circuit C into a set of unitaries {U}

Breakdown of steps

• There are two options to get the noisy gate basis {G}:
• Using quantum device information entails quantum process tomography for each native gate the computer can implement.
• E.g., for IBM, U1, U2, U3, CNOT
• Question: What if the noisy versions of these gates don’t form a basis?
• Assumptions on noise lead to analytic expressions in which {G} is written in terms of {U}.
• E.g., Temme et al example with depolarizing + Pauli noise

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Input or compute: Noisy basis {G}

Breakdown of steps

• Depending on how the noisy basis {G} is obtained, this step can be very simple or more involved.
• If assumptions about noise are made, there can be a simple rule for expressing {U} in terms of {G}
• E.g., for depolarizing + Pauli noise, the rule is to add a Pauli after each gate {U} with some probability [Temme et al].
• If quantum device information is used, we need a routine to solve the following linear program [Temme et al Appendix IV]:

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(2) Express each unitary in {U} in the basis of noisy gates {G}

Breakdown of steps

• This step is probably the most straightforward:
• Loop through the gates in C.
• Replace each gate U with a noisy basis gate G sampled according to its quasi-probability representation (QPR)
• The QPR is essentially the basis coefficients a_i of this gate U = a_1 G_1 + … + a_k G_k where {G} = {G_1, …, G_k}
• Execute this “noisy circuit” N using the input executor function E and store the result.
• Repeat this process M times if a finite number of samples is input. If a precision δ is input, repeat the process gamma_beta / δ^2 times where gamma_beta is the normalizing factor for the (total, not gate-wise) QPR.

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General PEC Pseudocode

Screenshot from https://github.com/unitaryfund/mitiq-internal/blob/master/pec/pec_prototype.ipynb.

See also https://github.com/unitaryfund/mitiq-internal/blob/master/pec/pec_draft.ipynb.