[论文解读] Testing symmetry on quantum computers

Margarite L. LaBorde, Soorya Rethinasamy+1 more|arXiv (Cornell University)|May 26, 2021

Quantum Computing Algorithms and Architecture被引用 7

一句话总结

本文提出利用量子计算机测试量子态与量子通道对称性的量子算法，证明这些算法的接受概率等于最大对称保真度，从而赋予这些度量以操作意义。该方法使用变分量子线路替代量子证明者，在无噪声与有噪声的模拟器中均表现出稳健性能，并应用于纠缠、可分性及不对称性资源理论。

ABSTRACT

Symmetry is a unifying concept in physics. In quantum information and beyond, it is known that quantum states possessing symmetry are not useful for certain information-processing tasks. For example, states that commute with a Hamiltonian realizing a time evolution are not useful for timekeeping during that evolution, and bipartite states that are highly extendible are not strongly entangled and thus not useful for basic tasks like teleportation. Motivated by this perspective, this paper details several quantum algorithms that test the symmetry of quantum states and channels. For the case of testing Bose symmetry of a state, we show that there is a simple and efficient quantum algorithm, while the tests for other kinds of symmetry rely on the aid of a quantum prover. We prove that the acceptance probability of each algorithm is equal to the maximum symmetric fidelity of the state being tested, thus giving a firm operational meaning to these latter resource quantifiers. Special cases of the algorithms test for incoherence or separability of quantum states. We evaluate the performance of these algorithms on choice examples by using the variational approach to quantum algorithms, replacing the quantum prover with a parameterized circuit. We demonstrate this approach for numerous examples using the IBM quantum noiseless and noisy simulators, and we observe that the algorithms perform well in the noiseless case and exhibit noise resilience in the noisy case. We also show that the maximum symmetric fidelities can be calculated by semi-definite programs, which is useful for benchmarking the performance of these algorithms for sufficiently small examples. Finally, we establish various generalizations of the resource theory of asymmetry, with the upshot being that the acceptance probabilities of the algorithms are resource monotones and thus well motivated from the resource-theoretic perspective.

研究动机与目标

• 开发用于测试量子态与量子通道对称性的量子算法，为对称保真度度量赋予操作意义。
• 将不对称性资源理论推广至先前工作的范围之外，包括玻色对称性与可扩展性。
• 利用变分量子算法实现在近期量子设备上对对称性的实用测试。
• 使用半定规划对小规模实例进行性能基准测试。
• 将框架扩展至量子通道，包括协方差对称性与不可 signaling 约束。

提出的方法

• 提出使用量子线路测试 $G$-玻色对称性、$G$-对称性及其可扩展版本的量子算法。
• 在互动证明框架中使用量子证明者，随后由参数化变分线路替代，以适配近期设备。
• 通过酉表示与对称投影，证明接受概率等于最大对称保真度。
• 采用半定规划（SDP）计算小例子的最大对称保真度，用于基准测试。
• 将变分方法应用于测试两体与多体态的 $k$-可扩展性与 $k$-玻色可扩展性。
• 推广不对称性资源理论，证明接受概率为资源单调量。

Figure 1: Quantum circuit to implement Algorithm 1 . The unitary $U^{\rho}$ prepares a purification $\psi_{S^{\prime}S}$ of the state $\rho_{S}$ . The final measurement box with the plus-sign to the right of it indicates that the measurement $\{|+\rangle\!\langle+|_{C},\mathbb{I}_{C}-|+\rangle\!\lan

实验结果

研究问题

• RQ1 能否设计出具有操作意义的量子算法，用于测试量子态与通道的对称性？
• RQ2 最大对称保真度如何与量子测试的接受概率关联？
• RQ3 变分量子线路能否有效替代近期量子硬件中对称性测试的量子证明者？
• RQ4 这些算法在有噪声与无噪声量子模拟器上的性能如何？
• RQ5 所提出的对称性概念如何推广现有概念，如可分性、不可扩展性与框架性？

主要发现

• 每个对称性测试算法的接受概率等于输入态的最大对称保真度，从而为这一资源度量赋予操作意义。
• 算法在无噪声模拟中表现良好，并在有噪声模拟器中展现出对噪声的鲁棒性，验证了其实际可行性。
• 半定规划可精确计算小例子的最大对称保真度，从而实现对量子算法的基准测试。
• 变分方法成功替代了量子证明者，使当前量子硬件能够估计对称保真度。
• 该框架推广了不对称性资源理论，证明接受概率为资源单调量，从而支持其作为度量的合理性。
• 该方法可扩展至量子通道，已提出用于测试协方差对称性与通道 $G$-对称可扩展性的算法。

Figure 2: Unitary $U_{d}$ , with $\theta=2\arctan\!\left(\frac{1}{\sqrt{2}}\right)$ , generates the equal superposition of six elements from ( 43 ). Note that the controlled-Hadamard is controlled on the qubit being in the state zero.