Symmetric quantum computation

TL;DR

It is established that symmetric quantum circuits are fundamentally more powerful than their classical counterparts.

Abstract

We introduce a systematic study of "symmetric quantum circuits", a new restricted model of quantum computation that preserves the symmetries of the problems it solves. This model is well-adapted for studying the role of symmetry in quantum speedups, extending a central notion of symmetric computation studied in the classical setting. Our results establish that symmetric quantum circuits are fundamentally more powerful than their classical counterparts. First, we give efficient symmetric circuits for key quantum techniques such as amplitude amplification, phase estimation and linear combination of unitaries. In addition, we show how the task of symmetric state preparation can be performed efficiently in several natural cases. Finally, we demonstrate an exponential separation in the symmetric setting for the problem XOR-SAT, which requires exponential-size symmetric classical circuits but can be solved by polynomial-size symmetric quantum circuits.

Figures (9)

• Figure 1: Two ways of concatenating circuits which use workspace qubits.
• Figure 2: Implementing a controlled single-qubit gate in five layers.
• Figure 3: Implementing a controlled threshold gate in three layers.
• Figure 4: Implementing the circuit $\ket{j}\!\bra{j} \otimes e^{i \theta_j} U_j + (I_{2^m} - \ket{j}\!\bra{j}) \otimes I_{2^{n}}$ using one extra ancilla and a controlled $\mathcal{C}_j$.
• Figure 5: The circuit for symmetric amplitude amplification, with the reflections $\mathcal{R}_G$ and $\mathcal{R}_S$ highlighted.

Theorems & Definitions (72)

• Theorem 1: Symmetric quantum subroutines
• Theorem 2: Symmetric state preparation
• Theorem 3: Symmetric subspace unitaries
• Theorem 4: Equivariant QNNs
• Theorem 5: Symmetric quantum advantage
• Remark
• Definition 1: Threshold circuit
• Definition 2: Circuit automorphism
• Definition 3: Symmetric threshold circuit
• Example 1: Triangle-freeness