Phase Estimation with Compressed Controlled Time Evolution

TL;DR

This work tackles the bottleneck of encoding controlled time evolution for translationally invariant, local Hamiltonians in quantum simulation. It introduces Translationally Invariant Compressed Control (TICC), a compression framework that leverages a Pauli-string insertion-based equivalence and a two-term additive cost to approximate controlled evolution with near-optimal circuit depth $O(t polylog(t N/ε))$ while reducing the control overhead to an additive factor. The authors provide detailed 1D and 2D benchmark results on spin models (e.g., TFIM and Heisenberg on square and triangular lattices), showing sub-percent ground-state energy errors and substantial gate-count reductions for Iterative Quantum Phase Estimation on noisy hardware emulators. This work suggests a practical pathway to run QPE and related algorithms on near-term devices and informs hardware-aware circuit design for scalable quantum simulation.

Abstract

Many optimally scaling quantum simulation algorithms employ controlled time evolution of the Hamiltonian, which is typically the major bottleneck for their efficient implementation. This work establishes a compression protocol for encoding the controlled time evolution operator of translationally invariant, local Hamiltonians into a quantum circuit. It achieves a near-optimal scaling in circuit depth $\mathcal{O}(t \text{ polylog}(t N/ε))$, while reducing the control overhead from a multiplicative to an additive factor. We report that this compression protocol enables the implementation of Iterative Quantum Phase Estimation with as few as 414 CNOT gates for a frustrated quantum spin system on a 6x6 triangular lattice and delivers ground state energy errors below 1% (with $\pm$ 1.5% variation, calculated with a hardware noise aware pipeline) on a 4x4 triangular lattice using the noisy emulator of the Quantinuum H2 trapped ion device.

Figures (8)

• Figure 1: Conceptual visualization of the proposed protocol. (a) Brickwall Ansatz of $N=4$ qubits, employed to approximate the time evolution dynamics of a local, translationally symmetric Hamiltonian in one dimension. The light-cone shaped trajectory illustrates the information propagation with a finite velocity $v_{\text{LR}}$. Gates, optimized on this Ansatz, can be reused to simulate time dynamics of larger systems, if the targeted evolution time does not exceed a $t_{\text{max}}=\mathcal{O}(N/v_{\text{LR}})$, because for $t<t_{\text{max}}$ the light cone doesn't expand enough to entangle sites $(X, Y)$ with distance $\space \mathclap{l}\space(X,Y) \ge N$. (b)Top: Iterative Quantum Phase Estimation circuit. By inserting identity (inverse phase gate) in the red block, one samples the real (imaginary) part of the phase $e^{-it \lambda}$. Bottom: Translationally Invariant Compressed Control (TICC) circuit that approximates the controlled time evolution for a chain of four qubits. In this circuit, green gates effectively flip the evolution direction, hence controlling these green gates with the ancilla implements the equivalence given in Eq. \ref{['eq:cU_equivalence']}. Crucially, the number of green layers does not scale with respect to the targeted evolution time. Number of red layers $\gamma$, scales near-optimally with respect to evolution time $t$. (c) Post processing after the Hadamard test. Top: renormalizing the estimated phase amplitude, amplifies statistical error bars by projection onto the unit circle, incorporating depolarizing noise effects. Bottom: fit of measured phases for several time points into a phase curve, from which we infer the phase angle estimation ($e^{-i t E_{\text{fit}}}$).
• Figure 2: Ansatz employed to optimize a TICC circuit of time $T$, for one dimensional, isotropic Heisenberg model (\ref{['eq:isotropic_HM']}). Subscripts and gate labels indicate the starting point of the optimization, and they correspond to a splitting with respect to the decomposition $H_1 = \sum_{\langle i,j \rangle} X_iX_j+Y_iY_j$ and $H_2 = \sum_{\langle i,j \rangle} Z_iZ_j$. This decomposition implements a splitting of the form given in Eq. \ref{['eq:ham_decomposition']} with $\eta =2$, $K_1 = Z \otimes I \otimes Z \dots$ and $K_2 = X \otimes I \otimes X \dots$. Because each red block employs $\gamma = 3$ layers, we start the optimization from an initial point that corresponds to implementing each $e^{-iT H_i}$ term with a second order Trotterization, where the gate labels in red blocks indicate the time steps $t$ used in each $e^{-i t H_i}$ term. Although we start from a separable state for the green layers, the Ansatz allows entanglement between sites $\{(2j, 2j+1)\}_{j=0}^1$.
• Figure 3: (a) Iterative QPE, employed for ground state energy estimations of the antiferromagnetic TFIM (\ref{['eq:TFIM']}) with various field strengths on a triangular 4×4 lattice. We start from the initial energy estimates at {-40, -45, -50, -60} for the field strengths considered $g \in \{1.5, 2, 2.5, 3\}$. These values were interpolated from the ground state energy solutions of 2x2 and 3x3 system sizes (which were solved by exact diagonalization) and then rounded up to the nearest multiple of 5. We run the iterative scheme (whose details are explained in App. \ref{['sec:A_QPE']}) for times $t=0.125$ and $t=0.25$ ($k \in \{-3, -2\}$). To encode the controlled time evolution operator, we employ two approaches: Second order Trotterization with Pauli string insertion, explained in Sec. \ref{['sec:sec2']} (blue) and TICC (green). TICC circuits employ 184 hardware-native ZZPhase gates per phase estimation circuit, whereas Trotter circuits employ 336. We use 200 shots per expectation value measurement for both methods. With all of the field strengths considered, TICC delivers sub-1% relative energy error ({0.7%, 0.1%, 0.2%, 0.2%} for the given transverse field strengths) with $\pm 1.5\%$ variation. These error bars are calculated through the protocol explained in App. \ref{['sec:A_QPE']} and visualized in Fig.\ref{['fig:fig1']}.c, with an average amplification factor of $\sim 1.6$ for the TICC results. We interpolate between the asymptotic linear behavior of the energy at large field strengths and the classical energy (at zero field) to qualitatively estimate the pseudo–phase transition point ($g_c$). (b) Correlation function at transverse field $g=2$(i) and (dis)order parameters $|\langle X\rangle|$ at $g\in\{1.5, 2, 2.5\}$(ii), sampled through QFT based QPE protocol for basis time $t_0=0.1$ with two ancilla qubits. Controlled time evolutions are implemented with TICC circuits. We use 2000 shots for expectation value measurement per each basis (X and Z) and the error bars represent the statistical, standard deviation. Each circuit employs 366 hardware native ZZPhase gates for the full, QFT appended protocol. Results in both plots are obtained from simulations on the noise-aware emulator of the Quantinuum H2 device, whose noise parameters are listed in App. \ref{['sec:A3']}.
• Figure A1: Two staged decomposition of the large unitary into smaller sub-unitaries, as given in Eqs. \ref{['eq:decomposition']} & \ref{['eq:decomposition2']}. For the case of translationally invariant $H_X$ under local translations: $H_{A \cup B} = H_{C}$ and $H_B = H_E$, hence the resulting circuit is translationally symmetric. This decomposition is the same as the sequence used in Haah_2021, with the assumption of $\norm{A \cup B} = \norm{C}$.
• Figure A2: Numerical simulations of circuit depth scaling of the RQC-opt protocol. $\epsilon$ is the spectral norm distance error of the approximate time evolution unitary and $t$ is the evolution time. We fit the data points to the function given in Eq. \ref{['eq:fit']} and report that the optimal parameter corresponding to $c_1$ is approximately 1, suggesting that RQC-opt satisfies the near-optimal scaling $\mathcal{O}(t \log(t/\epsilon))$ for the circuit depth.

Theorems & Definitions (2)

• Theorem 1: Existence of local approximants Haah_2021
• Theorem 2: Existence of translationally invariant, local approximants