Query-optimal estimation of unitary channels in diamond distance

TL;DR

The paper tackles the problem of learning an unknown $d$-dimensional unitary channel under the diamond-norm metric. It introduces a space-efficient, single-qudit algorithm that achieves $\varepsilon$-closeness with $O(d^2/\varepsilon)$ queries, and proves a matching lower bound $\Omega(d^2/\varepsilon)$, establishing optimality in both query and space complexity. The key innovations are a base tomography routine with $O(d^2/\varepsilon^2)$ calls, and a novel bootstrap that leverages unitary-group geometry to convert constant-error estimates into $\varepsilon$-error estimates with Heisenberg scaling, via a shift-to-identity technique and fractional-root refinements. The results substantially improve prior efforts (notably those achieving $O(d^3/\varepsilon^2)$ or $O(d^{2.5}/\varepsilon)$) and close the gap between practical, space-efficient tomography and information-theoretic limits, with implications for quantum programming and unitary learning. The paper also discusses extensions to eigenvalue estimation, gate-depth considerations, and the limitations when moving beyond unitary channels.

Abstract

We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a $\textsf{d}$-dimensional qudit, we aim to output a classical description of a unitary that is $\varepsilon$-close to the unknown unitary in diamond norm. We design an algorithm achieving error $\varepsilon$ using $O(\textsf{d}^2/\varepsilon)$ applications of the unknown channel and only one qudit. This improves over prior results, which use $O(\textsf{d}^3/\varepsilon^2)$ [via standard process tomography] or $O(\textsf{d}^{2.5}/\varepsilon)$ [Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to "bootstrap" an algorithm that can produce constant-error estimates to one that can produce $\varepsilon$-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires $Ω(\textsf{d}^2/\varepsilon)$ applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity.

Figures (2)

• Figure 1: The warmup example: we wish to learn $Z = (1\phi)$, where we do not know the phase $\phi$ on the complex unit circle. Our strategy is to get constant-error estimates to $Z^{2^k}$, which correspond to phases $\psi_k$ that specify a constant-sized interval around $\phi^{2^k}$ for each $k$. Although each interval only pins down the corresponding power of $\phi$ to constant error, each $\psi_k$ can be thought of as specifying the $k$th "bit" of information of $\phi$. So, when taken together, these error intervals can be collated to get an estimate of $\phi$ to error $O(\varepsilon)$ for the cost of computing constant-error estimates of powers of $Z$ up to $Z^{1/\varepsilon}$.
• Figure 2: The augmented version $\widehat{\mathop{\mathrm{\mathcal{C}}}\nolimits}$ of the circuit $\mathop{\mathrm{\mathcal{C}}}\nolimits$, which alternates between unitaries $U_i$ and an application of $Z$ or $Z^\dagger$, to perform $Q$ queries in total. Each application of $Z$ or $Z^\dagger$ is replaced with the gadget in \ref{['eq:fractional-power-circuit']}, making for $Q$ additional ancilla qubits. The dotted box indicates the state prepared in the ancilla, denoted $\ket{\textup{anc}}$ in \ref{['eq:anc-def']}. Figure taken from bccks17 with slight modifications.

Theorems & Definitions (60)

• theorem 1.1: Upper bound
• theorem 1.2: Lower bound
• definition 1.3
• definition 1.4
• definition 1.5
• Proposition 1.5: Equivalence of diamond norm, operator norm, intrinsic Lie metrics
• definition 1.6
• remark 1.7
• Proposition 1.7
• definition 1.8