Approximation does not help in quantum unitary time-reversal

TL;DR

This work proves a robust and tight lower bound for unitary time-reversal: any algorithm that approximates the inverse of an unknown $d$-dimensional unitary $U$ to a fixed error $\epsilon$ requires $\Omega((1-\epsilon)d^2)$ queries, even under adaptive, coherent strategies with unbounded ancillas. The authors model time-reversal as a quantum comb and analyze the Haar-averaged performance operator $\mathbb{E}_U[C_U]$ using a representation-theoretic framework built from Schur-Weyl duality, Young diagrams, and Kerov’s interlacing sequences. Central to the argument are the stair operators $\{A_k\}$, which bound the Haar moment in the Löwner order and contract with arbitrary combs under the link product, yielding a depolarizing-type constraint unless $n$ is of order $d^2$. The results extend to generalized time-reversal $U^{-t}$ and imply strong hardness even for low-depth or cryptographically constructed unitaries, thereby showing that approximation does not improve the query complexity and matching the known $O(d^2)$ upper bound for exact reversal. The findings have broad implications for quantum learning, metrology, and higher-order transformations, clarifying the landscape of robust hardness for quantum transformation tasks.

Abstract

Access to the time-reverse $U^{-1}$ of an unknown quantum unitary process $U$ is widely assumed in quantum learning, metrology, and many-body physics. The fundamental task of unitary time-reversal dictates implementing $U^{-1}$ to within diamond-norm error $ε$ using black-box queries to the $d$-dimensional unitary $U$. Although the query complexity of this task has been extensively studied, existing lower bounds either hold only for the exact case (i.e., $ε=0$) or are suboptimal in $d$. This raises a central question: does approximation help reduce the query complexity of unitary time-reversal? We settle this question in the negative by establishing a robust and tight lower bound $Ω((1-ε)d^2)$ with explicit dependence on the error $ε$. This implies that unitary time-reversal retains optimal exponential hardness (in the number of qubits) even when constant error is allowed. Our bound applies to adaptive and coherent algorithms with unbounded ancillas and holds even when $ε$ is an average-case distance error.

Figures (5)

• Figure 1: The combination of a $4$-comb $X$ with a $3$-comb $Y$, resulting in a $1$-comb $X\star Y$ on $(\mathcal{H}_0,\mathcal{H}_7)$.
• Figure 2: Rotated version of the Young diagram $\lambda=(6,5,3,3,2,1,1,1)$.
• Figure 3: The quantum comb $R$ is a unitary time-reversal algorithm with error bounded by $\epsilon$ using $n$ queries to $U$ and the overall channel from $\mathcal{H}_0$ to $\mathcal{H}_{2n+2}$ approximates the identity channel $\mathcal{I}$, i.e., $R\star C_U \stackrel{\epsilon}{\approx} |I\rangle\!\rangle\!\langle\!\langle I|$.
• Figure 4: Properties of the stair operators $\{A_k\}_{k=1}^n$ (see \ref{['lemma-641548']} and \ref{['lemma-641551']}).
• Figure 5: An example illustrating the idea in the proof of \ref{['lemma-6161632']}.

Theorems & Definitions (50)

• Theorem 1.2: \ref{['thm-640254']} and \ref{['coro-6290408']} restated
• Corollary 1.3: \ref{['cor:generalized-time-reversal']} restated
• Corollary 1.4
• proof
• Corollary 1.5
• Definition 2.1: Diamond norm AKN98
• Definition 2.2: Average-case distance huang2024learningvasconcelos2024learning
• Definition 2.3: Quantum comb chiribella2009theoretical
• Definition 2.4: Link product "$\star$" chiribella2009theoretical
• Remark 2.5