Analytical Lower Bound on Query Complexity for Transformations of Unknown Unitary Operations

TL;DR

This Letter establishes analytical lower bounds for the query complexity of unitary inversion, transposition, and complex conjugation, which hold even if the input unitary is an unknown logarithmic-depth unitary.

Abstract

Recent developments have revealed deterministic and exact protocols for performing complex conjugation, inversion, and transposition of a general $d$-dimensional unknown unitary operation using a finite number of queries to a black-box unitary operation. In this work, we establish analytical lower bounds for the query complexity of unitary inversion, transposition, and complex conjugation, which hold even if the input unitary is an unknown logarithmic-depth unitary. Specifically, our lower bound of $d^2$ for unitary inversion demonstrates the asymptotic optimality of the deterministic exact inversion protocol, which operates with $O(d^2)$ queries. We introduce a novel framework utilizing differentiation to derive these lower bounds on query complexity for general differentiable functions $f: \mathrm{SU}(d)\to \mathrm{SU}(d)$. As a corollary, we prove that a catalytic protocol -- a new concept recently noted in the study of exact unitary inversion -- is impossible for unitary complex conjugation. Furthermore, we extend our framework to the partially known setting, where the input unitary operation is promised to be within a subgroup of $\mathrm{SU}(d)$ and the probabilistic setting, where transformations succeed probabilistically.

Figures (7)

• Figure 1: The quantum circuit implementing deterministic and exact transformation $f(U)$ for a black-box unitary operation $U$ with $N$ queries to $U$, where $\rho_A$ is a fixed state of the auxiliary system, and $V_1,\ldots ,V_{N+1}$ are unitary operations. $Z(U)$ is the unitary operation corresponding to the circuit without $\rho_A$ and tracing out.
• Figure 2: Summary of upper ("Our upper bound") and lower (other lines) bounds of the success probability of unitary transposition. "Our upper bound" shows the analytical solution of the SDP for the probabilistic transformation. "Success or draw" and "Parallel" refer to the lower bounds corresponding to the success probability of protocols given in Sec. E and Thm. 2 of quintino2019probabilistic, respectively. Magenta star shows the number of queries $\sim (\pi /2)d^2$ required in the deterministic exact transposition algorithm given by modifying the algorithm in chen2024quantum.
• Figure S1: Quantum circuit implementing $f(U)$ using $N$ queries of an unknown unitary operation $U$. The upper and lower lines represent the main system $\mathcal{H}$ and the auxiliary system $\mathcal{H}_A$, respectively, $\rho_A$ is a quantum state of the auxiliary system, and $V_1,\ldots ,V_{N+1}$ are unitary operators on the composite system. $Z(U)$ is the unitary operation corresponding to the circuit excluding $\rho_A$ and tracing out.
• Figure S2: Quantum circuit in Fig. \ref{['fig::lowerbound_comb']} is transformed to this circuit in order to simplify the expression in the neighborhood of $U=U_0$.
• Figure :

Theorems & Definitions (15)

• Theorem 1
• proof : Proof sketch
• Corollary 2
• proof : Proof sketch
• Corollary 3
• Theorem 4
• proof : Proof
• Lemma S1
• Lemma S2
• Theorem S5