Quantum Simulation of Random Unitaries from Clebsch-Gordan Transforms

TL;DR

This construction can be viewed as a representation-theoretic generalization of Zhandry's compressed function oracle technique and utilizes Clebsch-Gordan transforms as main building blocks for the unitary group.

Abstract

We present a general method for simulating an action of $t$ copies of a Haar random unitary for arbitrary compact groups. This construction can be viewed as a representation-theoretic generalization of Zhandry's compressed function oracle technique. It is conceptually simple, exact and utilizes Clebsch-Gordan transforms as main building blocks. In particular, for the unitary group, our method is efficient in space and time. Finally, our general oracle for forward queries can be easily modified into oracles for conjugate, transpose, and inverse queries, thus unifying all four query types.

Figures (10)

• Figure 1: (a) A quantum circuit involving $t$ queries to a unitary operations $U,\bar{U},U^{\mathsf{T}}$ and $U^{\dagger}$, where $U$ is drawn from the Haar measure. The boxes other than $U$ represent arbitrary quantum channels. (b) Unitary $t$-design $\{p_i, U_i\}_i$ can simulate the quantum circuit (a) by using $U_i$ with probability $p_i$. (c) Our construction simulates the quantum circuit (a) exactly by using the compressed oracles $\mathrm{fO},\mathrm{cO},\mathrm{tO},\mathrm{iO}$ and tracing out the auxiliary register.
• Figure 2: Compressed oracle $\mathrm{fO}$ for an arbitrary group $G$ written in standard quantum circuit notation (time goes from left to right). $x_k$ denotes the input, $y_k$---the output, while $M$ and $\bar{M}$ label basis states of the ancilla registers, which store irreducible representations of $G$. Clebsch--Gordan transform decomposes tensor product of representations $\lambda \otimes R$, where $\lambda$ is some irrep and $R$ is a given unitary representation of $G$. Such decomposition is in general not multiplicity-free, and the register carrying the multiplicity is highlighted with a slash wire.
• Figure 3: Four types of oracles based on the corresponding query types: forward $\mathrm{fO}$, conjugate $\mathrm{cO}$, transpose $\mathrm{tO}$, inverse $\mathrm{iO}$.
• Figure 4: Proof idea behind the construction of our compressed oracle $\mathrm{fO}$. The top figure is drawn in the Heisenberg picture and represents $t$ queries to the oracle $\mathrm{fO}$. The middle figure is the rewriting of the top one in terms of (dual) Clebsch--Gordan tensors, which comprise matrix units of the commutant of $R^{\otimes t}$ action. The bottom is an equivalent Haar integral expression. The equalities are proven in detail in Appendix \ref{['sec:app_proof_main']} of the SM supple.
• Figure 5: Twirling of the approximate unitary inversion protocol using the forward compressed oracle $\mathrm{fO}$. The top figure corresponds to a quantum comb that approximately implements unitary inversion with $n$ queries to the input unitary channel $U$ with the average-case channel fidelity $F = 1-\delta$ defined in Eq. \ref{['eq:fidelity']}. The bottom figure corresponds to the twirled quantum comb that transforms $n$ queries of $U$ into the channel given by $\mathcal{D}_{\eta(\delta)} \circ \mathcal{U}^{-1}$, where $\mathcal{D}_{\eta(\delta)}$ is a depolarizing channel with the noise parameter given by $\eta(\delta) = {d^2 \over d^2-1} \delta$.

Theorems & Definitions (9)

• Theorem 1: Informal
• proof : Proof idea
• Theorem 2
• proof : Proof idea
• Lemma 3
• proof
• Theorem 4
• proof
• Remark 5