Optimal Hamiltonian recognition of unknown quantum dynamics

TL;DR

The paper defines Hamiltonian recognition as identifying an unknown Hamiltonian from a known set by querying the induced unitary $U_H(\theta)$, linking quantum channel discrimination with quantum metrology. It develops a framework combining quantum testers and quantum signal processing (QSP) to construct optimal recognition protocols, proving that the average error decays as $O(1/k)$ with the number of queries $k$. For binary recognition with orthogonal Pauli directions, the authors derive the exact optimal success probability $\mathrm{Suc}_k^{GEN}=\frac{k+\max\{p,1-p\}}{k+1}$ and implement a QSP-based, entanglement-free protocol that saturates this bound; they extend the approach to ternary recognition with $\{X,Y,Z\}$, achieving $\mathrm{Suc}_k^{SEQ}=\frac{k+\max\{p_0,p_1,p_2\}}{k+1}$ for odd $k$ using two coherent QSP circuits. They further generalize to arbitrary axis directions via rotations and provide numerical SDP evidence for multi-qubit Heisenberg-type Hamiltonians, along with experimental demonstrations on superconducting hardware. Overall, the work offers an efficient, time-independent method to recognize Hamiltonians from limited dynamics, bridging composite channel discrimination and quantum metrology, and suggesting new directions for quantum algorithm design and verification.

Abstract

Identifying unknown Hamiltonians from their quantum dynamics is a pivotal challenge in quantum technologies. In this paper, we introduce Hamiltonian recognition, a framework that bridges quantum hypothesis testing and quantum metrology, aiming to identify the Hamiltonian governing quantum dynamics from a known set of Hamiltonians. To identify $H$ for an unknown qubit quantum evolution $\exp(-iHθ)$ with unknown $θ$, from two or three orthogonal Hamiltonians, we develop a quantum algorithm for coherent function simulation, built on two quantum signal processing (QSP) structures. It can simultaneously realize a target polynomial based on measurement results regardless of the chosen signal unitary for the QSP. Utilizing semidefinite optimization and group representation theory, we prove that our methods achieve the optimal average success probability, taken over possible Hamiltonians $H$ and parameters $θ$, decays as $O(1/k)$ with $k$ queries of the unknown unitary transformation. Furthermore, we demonstrate the validity of our protocol on a superconducting quantum processor. We also investigate a physically motivated recognition task for Heisenberg Hamiltonians, providing numerical evidence for effective multi-qubit quantum system recognition. This work presents an efficient method to recognize Hamiltonians from limited queries of the dynamics, opening new avenues in composite channel discrimination and quantum metrology.

Figures (6)

• Figure 1: Optimal protocols for the recognition of $\{X,Z\}$. The protocols consist of rotation gates around the $x$-axis (in pink) and the $z$-axis (in purple). The top circuit illustrates the protocol for an odd number of queries, while the bottom circuit represents the protocol for an even number of queries.
• Figure 2: Performance of binary Hamiltonian recognition for ${X, Z}$. (a) The success probability of recognizing $X$ by querying an unknown unitary $k$ times. The $x$-axis represents the parameter $\theta$ for the unknown unitary, and the $y$-axis corresponds to the success probability of recognizing $X$. (b) Experiment results of the recognition protocol on a surface-13 tunable coupling superconducting quantum processor with different numbers of queries to the unknown unitary. In each panel, the $y$-axis corresponds to $\theta$ for $e^{-iH\theta}$, where the blue points represent unitaries generated by the Hamiltonian $H=Z$ and the orange points represent unitaries generated by $H=X$. For each value of $\theta$, 512 points of each color are shown, corresponding to 1024 repeated experiments in total. The left region corresponds to unitaries generated by $Z$, and the right region to those generated by $X$. Increasing the number of queries reduces the misclassification of points between the two regions.
• Figure 3: The three-qubit quantum circuit for ternary Hamiltonian recognition of $\{X,Y,Z\}$. The input state is $|000\rangle$. If the measurement outcome is $s_0s_1 = 00$ or $10$, determine $H=Z$. If the measurement outcome is $s_0s_1 = 01$, determine $H=Y$. If the outcome is $s_0s_1 = 11$, determine $H=X$.
• Figure 4: Average success probability of recognizing $\{H_0, Z\}$. The plots illustrate the performance for query counts $k=1$ (left) and $k=3$ (right) as a function of the vector components $n_{0,x}$ and $n_{0,z}$ of the Hamiltonian $H_0$. In each panel, three performance layers are compared: the top surface represents the globally optimal success probability derived via semidefinite programming (SDP); the middle surface depicts the performance of the proposed ${X, Z}$ QSP protocol applied to this general case; and the bottom surface represents the random guessing baseline.
• Figure 5: Average success probability for recognizing (a) two-qubit Hamiltonians $J_x X_1 X_2 + J_y Y_1 Y_2$ and (b) three-qubit Hamiltonians $J_x (X_1 X_2+X_2X_3) + J_y (Y_1 Y_2 + Y_2 Y_3)$. Each curve represents a different pair of Hamiltonians $\{\hat{H}_0, \hat{H}_1\}$, indexed by the parameter $(J_x,J_y)$ and $(-J_x, -J_y)$. The $x$-axis represents the number of queries to the unknown unitary, and the $y$-axis represents the optimized average success probability. It shows that the average success probability of Hamiltonian recognition increases with the number of queries, $k$, for all instances of Hamiltonian pairs considered.

Theorems & Definitions (16)

• Definition 1
• Theorem 1
• Corollary 2
• Theorem 3
• Theorem S1: gilyen_quantum_2018, Theorem 4
• Theorem S2: Martyn_2021, Theorem 9
• Lemma S3
• proof
• Lemma S4
• proof