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Error mitigation + error correction

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March 15nd, 2022 - Unitary Fund Group Meeting

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Andrea Mari

Special focus on: �Error mitigation for universal gates on encoded qubits,

Christophe Piveteau, David Sutter, Sergey Bravyi, Jay M. Gambetta, Kristan Temme

�https://arxiv.org/abs/2103.04915

Outline

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• The universal gate set {H, S, CNOT, T}.

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• Error correction

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• Magic states

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• Main idea and results of Piveteau et al.

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< H, S, CNOT > / U(1)

S

S

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Magic States

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{ H, S, CNOT} augmented with the T gate is a universal gateset

Hard to implement in a fault-tolerant way

Outline

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• The universal gate set {H, S, CNOT, T}.

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• Error correction

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• Magic states

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• Main idea and results of Piveteau et al.

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Phys.�gate

Quantum error correction

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• Encode logical qubits into many physical qubits ==> Fault-tolerant computation

• Two important types of codes: stabilizer codes, surface code.

Clifford gates can be

applied transversally

Clifford gates can be applied transversally apart from the S gate

Transversal gate

Logical qubits

=

Phys.�gate

Phys.�gate

Physical qubits

T gates cannot be applied transversally ⇒ T gates are very hard to error correct!

Outline

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• The universal gate set {H, S, CNOT, T}.

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• Error correction

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• Magic states

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• Main idea and results of Piveteau et al.

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Magic States

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Magic states are states |A> such that:

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Adaptive Clifford operations applied to |0>...|0> |A> ….|A> is universal.

Example of a magic state:

Different notations for the same state, both used in this talk.

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T-gate gadget: first example

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T-gate gadget: second example

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Outline

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• The universal gate set {H, S, CNOT, T}.

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• Error correction

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• Magic states

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• Main idea and results of Piveteau et al.

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Instead of the ideal magic state we can only prepare a noisy state

Twirling with respect to:

Error rate

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What channel do we get if we replace �the ideal magic state

with the noisy magic state ?

Result = a T gate with dephasing noise in the logical space

• Main idea of this work: Can we apply error mitigation (PEC) to noisy T gates?
• Main result of this work: How can we efficiently apply PEC in the logical space?

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• Main idea of this work: Can we apply error mitigation (PEC) to noisy T gates?

Ideal T gate = linear combination of noisy operations

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• Main result of this work: How can we efficiently apply PEC in the logical space?

First method:�

• Encode |T> states such that the logical error rate scales linearly with physical error rate.

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• Apply PEC as usual in the logical space.

Sampling overhead:

Physical error rate

A constant

Logical error rate

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• Main result of this work: How can we efficiently apply PEC in the logical space?

Second method (not considered in this talk).�

• Use code switching error correction method. Based on two encodings S1 and S2.

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• Code switching can be used such that

T logical gate = T physical gate on a single qubit

Disadvantage = larger PEC overhead. ~50 times more.

Advantage = T-gate gadgets are not necessary (no auxiliary qubits)

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Results

Number of T gates

Physical error rate

Assume k=2/5

2000 T gates

200 T gates

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Conclusions

• Error mitigation can be applied to noisy logical T gates before full fault tolerance is achieved.

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• This approach works only for algorithms in which the output are expectation values.

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• What can we do with Mitiq?

�Error mitigation for universal gates on encoded qubits,

Christophe Piveteau, David Sutter, Sergey Bravyi, Jay M. Gambetta, Kristan Temme

https://arxiv.org/abs/2103.04915