Quantum Krylov Subspace Diagonalization via Time Reversal Symmetries

TL;DR

The paper tackles the challenge of performing Krylov subspace diagonalization for near-term quantum devices by exploiting time-reversal symmetry. It introduces Krylov Time Reversal (KTR), which relies on a Hermitian involutory time-reversal operator $T$ with $\{\mathcal{H}, T\}=0$ and an initial state $T|v_0\rangle=c|v_0\rangle$, to obtain real-valued overlaps and reduce circuit depth. Theoretical results show that inner products $\langle v(t_a)|v(t_b)\rangle$ collapse to real expectation values $\langle v(h)|T|v(h)\rangle$ and that overlaps involving a commuting Hermitian $K$ reduce to expectations of $iKT$, enabling efficient, low-control measurements. The method includes an explicit initial-state construction via projections and block-structured circuits, with connections to the implicit Hadamard test, and is validated through tensor-network simulations on TFIM and a $\mathbb{Z}_2$ gauge Higgs model, achieving accurate spectral estimates with shallow circuits. Overall, KTR offers a practical route to quantum spectral estimation on noisy, shallow hardware while preserving favorable convergence properties of Krylov diagonalization, with potential applicability to gauge theories and other time-reversal symmetric Hamiltonians.

Abstract

Krylov quantum diagonalization methods have emerged as a promising use case for quantum computers. However, many existing implementations rely on controlled operations, which pose challenges to near-term quantum hardware. We introduce a novel protocol, termed Krylov Time Reversal (KTR), that circumvents these bottlenecks by leveraging time-reversal symmetry in Hamiltonian evolution. Using symmetric time dynamics, we show that it is possible to recover real-valued Krylov matrix elements, which significantly reduces the circuit depth and enhances compatibility with shallow quantum architectures. Furthermore, the protocol's structure indirectly reduces the total evolution time, benefiting both near-term and long-term architectures. We validate our method through numerical simulations on paradigmatic Hamiltonians exhibiting time-reversal symmetry, including the transverse-field Ising model and a lattice gauge theory, demonstrating accurate spectral estimation and favorable circuit constructions.

Figures (7)

• Figure 1: Block circuit (case $r=4$ qubits) for the preparation of the state $-\ket{w_0}$ in \ref{['eq:ising-init-state-block-i']}. The product $\ket{v_0}=\ket{w_0}^{\otimes s}$ yields the initial state.
• Figure 2: Relative error of the ground energy $\lambda_0$ for the TFIM on $n=64$ qubits with $m=128$ (MPS) Krylov vectors and $s$ the number of blocks $\ket{w_0}$. References for $\widehat{\lambda}_0$ are KQD ($\bullet$) and DMRG ($\blacktriangle$), with initial state $\ket{\varphi}=\ket{+}^{\otimes n}$.
• Figure 3: Relative error for the $\mathbb{Z}_2$ gauge Higgs model on $n=64$ qubits, relative to the ground energy for the sector $G\ket{\psi}=\ket{\psi}$. We considered $m=80$ Krylov vectors (MPS) and $s=2$ blocks $\ket{w_0}$.
• Figure 4: Numerical demonstration (MPS) of \ref{['thm:gram-hadamard-test-implicit']} and \ref{['thm:H-overlap-hadamard-test-implicit']} (implicit Hadamard test) for the TFIM on 64 qubits, reflecting the case of \ref{['section:example-ising']} and \ref{['fig:experiment-tfim']}. (a) Relative errors for the overlap matrices $A$ and $B$, where reference quantities are computed using the RHSs of \ref{['eq:gram-hadamard-test-implicit']} and \ref{['eq:H-overlap-hadamard-test-implicit']}. (b) Relative error of the ground energy $\lambda_0$ computed using the overlaps on the LHSs of \ref{['eq:gram-hadamard-test-implicit']} and \ref{['eq:H-overlap-hadamard-test-implicit']}. As references, we consider the canonical KQD ($\bullet$) and DMRG ($\blacktriangle$).
• Figure 5: General case for the projection $P_i$ applied to the arbitrary state $\ket{\varphi}$ (on $n=6$ qubits, with $s=2$ blocks) when the time reversal operator is of the form $T=\bigotimes_{l=1}^n V_l$, where $V_l$ is a non-identity, single-qubit involutory unitary (e.g. Paulis). This pattern can be extended to any number of blocks, with each block requiring an additional qubit. The unitaries $U_l$ satisfy the equation $U_l \sigma^{\mathrm{x}}_{} U_l^{\dagger}=V_l$, where $V_l$ is the target unitary (with eigenvalues $\{\pm 1\}$). The bits $i_1, i_2$ define the index $i=2 i_1 + i_2$ determining the projection $P_i$ in \ref{['eq:localized-ith-proj-struct']}. The coefficients $\xi_i \in (0, +\infty]$ are defined such that $\sum_i 1/\xi_i^2=1$. We note that the state is defined when $\xi_i$ is finite, that is, the probability $1/\xi_i^2$ is non-zero.

Theorems & Definitions (18)

• Lemma 3.1
• Lemma 3.2
• proof : Proof of \ref{['lemma:gram-from-expectation-on-T']}
• proof : Proof of \ref{['lemma:overlap-with-K-from-expectation-on-KT']}
• Lemma B.1
• Lemma C.1
• proof
• Theorem C.1
• proof
• Theorem C.2