Nonpublic Projects - Simple Encoding Scheme for Exponential Operators

Here is an introduction of my contribution to the project “Protection of Exponential Operation using Stabilizer Code in the Early Fault Tolerance Era”. This paper will be public on arXiv after internal review.

Here, the non-Clifford operators $\exp(-{\rm i}\theta P)$ is defined as exponential operator, exponential map or multi-qubit rotation, where $P$ is arbitrary $n$-qubit Pauli operator.

I. Abstract of the project

Quantum error correction offers a promising path to suppress errors in current quantum processors, but the resource required to protect logical operations from noise, especially non-Clifford operations, poses a substantial challenge to achieve practical quantum advantage in the early fault-tolerant quantum computing (EFTQC) era. In this work, we develop a systematic scheme to encode exponential map $\exp(-{\rm i}\theta P)$ into stabilizer codes with simple circuit structures and low qubit overhead. We then provide encoded circuits with small first-order logical error when performing post-selection using the $[[n, n-2, 2]]$ quantum error detecting code and the $[[5, 1, 3]]$, $[[7, 1, 3]]$, and $[[15, 7, 3]]$ quantum error correcting codes. Detailed analysis shows that under the level of physical noise of current devices, our encoding scheme is $4{-}7$ less noisy than the unencoded operation while only at most $3\%$ of runs need to be discarded.

II. Summary of my contribution

I developed a simple but useful encoding scheme of non-Clifford operators $\exp(-{\rm i}\theta P)$. This encoding scheme does not require compilation into universal set such as Clifford+$T$ thus avoid circuit synthesis errors due to compilation approximation. I also prove that any encoding scheme that directly implements encoded exponential map cannot be first-order fault-tolerant. In this context, I search for encoded circuit with minimum logical error rate for two-qubit or multi-qubit rotation in the $[[n, n-2, 2]]$ quantum error detecting code, the $[[5, 1, 3]]$, $[[7, 1, 3]]$, and $[[15, 7, 3]]$ quantum error correcting codes.

III. The idea of simple encoding

Motivation from EFTQC application. For early fault tolerant quantum computing, it is preferable to encode logical non-Clifford operations (especally non-Clifford operations) using relatively simple circuits and small stabilizer codes; these may not be completely fault tolerant but can still reduce noise effectively (see e.g., [1], [2], [3]).

Motivation from EFTQC algorithms. The exponential maps are the most essential non-Clifford building blocks of many algorithms which rely on variational ansätze of this form or trotterization of time evolution operators. It would be beneficial if one could protect the exponential map as a whole without Clifford+$T$ decomposition and magic state distillation.

IV. The encoding scheme and fault-tolerance

A general exponential map can be encoded using an arbitrary $[[n, k, d]]$ stabilizer code via

\[\overline{e^{-{\rm i}\theta P}} \xrightarrow{\text{ encoded into }} \exp(-{\rm i}{\theta}P),\]

where $\overline{e^{-{\rm i}\theta P}} = \cos \theta I - {\rm i}\sin\theta \overline{P} = e^{-{\rm i}\theta\overline{P}}$ and $P$ is a physical representative of the logical Pauli $\overline{P}$. We can prove that above encoding scheme does not change the codespace (i.e., commute with stabilizer generators).

However, any circuit that directly implements the encoded operator $e^{-{\rm i}\theta P}$ (i.e., using Clifford+$R(\theta)$ as universal gate set) is not first-order fault tolerant, since an over- or under-rotation of the encoded operator will turn into a logical error $e^{-{\rm i}\Delta \theta P}$. In this sense, circuits that minimize the first-order logical error for the encoded exponential map (referred to as optimal circuits) should ensure that only such imprecise rotations will introduce a logical error while all other physical errors are reliably detected by the syndrome measurement process.

V. Optimal circuit for $[[n, n-2, 2]]$ error detecting code

The optimal circuit for logical weight-$w$ operator $\overline{\exp(-{\rm i}\theta Z^{\otimes w}/2)}$ is given by

This circuit has one ancilla. Adding multiple ancilla does not improve the level of logical noise at least in my test. This circuit reduces logical error rate by $5-7$ times if $\rm CNOT$ depolarizing error rate is $0.001$ and $R_z(\theta)$ gate error is $0.001$ and up to $13$ times if $R_z(\theta)$ gate error is $0.0001$.

VI. Optimal circuit for $[[5,1,3]]$

The optimal circuit for logical three-qubit rotation $\overline{R_{zzz}(\theta)}$ is given by

This circuit has no ancilla and adding ancilla does not improve the level of logical noise. It is worth pointing out that logical $\rm CNOT$ of this code is not transveral gates. This circuit reduces logical error rate by $2-7$ times if $\rm CNOT$ depolarizing error rate is $0.001$ and $R_z(\theta)$ gate error is $0.001$ and up to $33$ times if $R_z(\theta)$ gate error is $0.0001$.

VII. Optimal circuit for $[[7,1,3]]$

The optimal circuit for logical three-qubit rotation $\overline{R_{zzz}(\theta)}$ is given by

This circuit has no ancilla and adding ancilla does not improve the level of logical noise.

VIII. Others

Any encoding weight-$w$ exponential operators can be obtained by sandwiching above circuits with single-qubit gates such as Hadamard gate $H$ and rotation gates $R_{x}(\pm \pi/2)$ or $R_{z}(\pm \pi/2)$, which does not change the level of logical noise.

It is worth mentioning that even for error correcting code, when computing the logical error rate, I assume that one perform post-selection rather than error correction when any of syndrome measurement indicates an error (i.e., any of measurements gets $-1$). This prevent the issue of introducing coherent error after applying correction. The number of discarded shots is not terribly large; instead, only a very small fraction of shots will be discarded, and noisy shots become smaller while physical noise level becomes smaller.

I do not show the optimal circuit for $[[15,7,3]]$ code for simplicity.