We give an algorithm that efficiently learns a quantum state prepared by
Clifford gates and $O(\log(n))$ non-Clifford gates. Specifically, for an
$n$-qubit state $\lvert \psi \rangle$ prepared with at most $t$ non-Clifford
gates, we show that $\mathsf{poly}(n,2^t,1/\epsilon)$ time and copies of
$\lvert \psi \rangle$ suffice to learn $\lvert \psi \rangle$ to trace distance
at most $\epsilon$. This result follows as a special case of an algorithm for
learning states with large stabilizer dimension, where a quantum state has
stabilizer dimension $k$ if it is stabilized by an abelian group of $2^k$ Pauli
operators. We also develop an efficient property testing algorithm for
stabilizer dimension, which may be of independent interest.

本研究报告提出了一种有效的算法，可以通过使用 Clifford 门和 O (log (n)) 非 Clifford 门来高效地学习一个量子态，并介绍了一种用于学习大稳定子维度的算法，其中量子态的稳定子维度使用一个由 2^k 个 Pauli 操作符的交换群来稳定。此外，我们还开发了一种有效的质检算法来测试稳定子维度。

利用少量非 Clifford 门高效学习准备好的量子态

Efficient Learning of Quantum States Prepared With Few Non-Clifford  Gates

Achieving near-term quantum advantage will require accurate estimation of
quantum observables despite significant hardware noise. For this purpose, we
propose a novel, scalable error-mitigation method that applies to gate-based
quantum computers. The method generates training data
$\{X_i^{\text{noisy}},X_i^{\text{exact}}\}$ via quantum circuits composed
largely of Clifford gates, which can be efficiently simulated classically,
where $X_i^{\text{noisy}}$ and $X_i^{\text{exact}}$ are noisy and noiseless
observables respectively. Fitting a linear ansatz to this data then allows for
the prediction of noise-free observables for arbitrary circuits. We analyze the
performance of our method versus the number of qubits, circuit depth, and
number of non-Clifford gates. We obtain an order-of-magnitude error reduction
for a ground-state energy problem on 16 qubits in an IBMQ quantum computer and
on a 64-qubit noisy simulator.

本文提出了一种新颖的、可扩展的误差缓解方法，旨在实现在存在硬件噪声的情况下准确估计量子观测值，该方法基于 Clifford 门，通过产生训练数据和使用线性 ansatz，使得可以预测任意电路的无噪声的观测值，运行在 16 个 qubit 的 IBMQ 量子计算机和 64 个 qubit 的噪声模拟器上，使用该方法可以将能量估计误差降至数量级。