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arXiv:2511.21225 · Physical Review A

Phase Estimation with Compressed Controlled Time Evolution

Erenay Karacan

quant-phSubmitted Nov 26, 2025• Announced Nov 27, 2025• 1 citation• Revised Apr 8, 2026 (v2)
v1 · Nov 26, 2025v2 · Apr 8, 2026
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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 in time $t$ scaling for 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 6$\times$6 triangular lattice and delivers ground state energy errors below 1% (with $\pm$ 1.5% variation, calculated with a hardware noise aware pipeline) on a 4$\times$4 triangular lattice using the noisy emulator of the Quantinuum H2 trapped ion device.

Journal reference
Phys. Rev. A 113, 042420 (2026)
DOI
10.1103/j26w-w48j

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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 in time ttt scaling for circuit depth O(t polylog(tN/ε))\mathcal{O}(t \text{ polylog}(t N/ε))O(t polylog(tN/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 6×\times×6 triangular lattice and delivers ground state energy errors below 1% (with ±\pm± 1.5% variation, calculated with a hardware noise aware pipeline) on a 4×\times×4 triangular lattice using the noisy emulator of the Quantinuum H2 trapped ion device.
Variational compression can significantly lower implementation overheads for encoding the time evolution of Hamiltonians into quantum circuits. However, they usually lack global convergence guarantees and well-established scaling behavior. In this work, we provide a recipe for choosing the initial point of such variational optimizations that guarantees convergence to a quantum circuit with near-optimal gate complexity O(N t polylog(N t/ε))\mathcal{O}\left( N \, t \, \text{polylog}(N \, t/ε) \right)O(Ntpolylog(Nt/ε)) for all local and translationally invariant Hamiltonians. We demonstrate our method by encoding the globally controlled time evolution of a Heisenberg antiferromagnet on a Kagome lattice. For N=48N = 48N=48 sites, evolution time t=0.1t=0.1t=0.1 and infidelity ε≈1%ε\approx1\%ε≈1%, the controlled time-evolution circuit requires 960 two-qubit B gates, for which we propose a straightforward implementation scheme for ion-trap setups. Thereby, our recipe extends digital quantum simulators toward system sizes and geometries that are challenging for classical computation.
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The time evolution of quantum many-body systems is one of the most promising applications for near-term quantum computers. However, the utility of current quantum devices is strongly hampered by the proliferation of hardware errors. The minimization of the circuit depth for a given quantum algorithm is therefore highly desirable, since shallow circuits generally are less vulnerable to decoherence. Recently, it was shown that variational circuits are a promising approach to outperform current state-of-the-art methods such as Trotter decomposition, although the optimal choice of parameters is a computationally demanding task. In this work, we demonstrate a simplification of the variational optimization of circuits implementing the time evolution operator of local Hamiltonians by directly encoding constraints of the physical system under consideration. We study the expressibility of such constrained variational circuits for different models and constraints. Our results show that the encoding of constraints allows a reduction of optimization cost by more than one order of magnitude and scalability to arbitrary large system sizes, without loosing accuracy in most systems. Furthermore, we discuss the exceptions in locally constrained systems and provide an explanation by means of an restricted lightcone width after incorporating the constraints into the circuits.
We present a hardware demonstration and resource analysis of split-evolution quantum phase estimation (SE-QPE) on a Quantinuum System Model H2 quantum computer. SE-QPE is a modification to canonical QPE for particle-conserving Hamiltonians in which controlled time evolution is replaced by CSWAP-based interference between a target register and a reference register. For factorizations of time evolution with a shared eigenbasis, SE-QPE preserves the phase-register outcome distribution of canonical QPE and, unlike with compute--uncompute substitutions, it remains compatible with non-exact eigenstates. The substitution removes controlled-simulation overhead and enables parallel evolution on two registers, reducing the depth of each phase-kickback block. Resource analysis for Trotterized double-factorized chemistry Hamiltonians shows that the substitution becomes increasingly favorable at higher phase powers and combining QPE and SE-QPE implementations can be a useful option. Over a range of FeMoco active spaces, SE-QPE reduces time evolution resources, with asymptotic reductions of about 33% in CX count, 25% in TTT count, and an asymptotic depth ratio of 3/N3/N3/N for CX layers. On Quantinuum H2-2, a four-qubit model ethylene demonstration with explicit inverse QFT and repeated phase-kickback steps up to 8 phase bits yields distinct energies and shows the auxiliary registers provide useful error detection filters.
We propose a phase-difference estimation algorithm based on the tensor-network circuit compression, leveraging time-evolution data to pursue scalability and higher accuracy on a quantum phase estimation (QPE)-type algorithm. Using tensor networks, we construct circuits composed solely of nearest-neighbor gates and extract time-evolution data by four-type circuit measurements. In addition, to enhance the accuracy of time-evolution and state-preparation circuits, we propose techniques based on algorithmic error mitigation and on iterative circuit optimization combined with merging into matrix product states, respectively. Verifications using a noiseless simulator for the 8-qubit one-dimensional Hubbard model using an ancilla qubit show that the proposed algorithm achieves accuracies with 0.4--4.7\% error from a true energy gap on an appropriate time-step size, and that accuracy improvements due to the algorithmic error mitigation are observed. We also confirm the enhancement of the overlap with matrix product states through iterative optimization. Finally, the proposed algorithm is demonstrated on IBM Heron devices with Q-CTRL error suppression for 8-, 36-, and 52-qubit models using more than 4,000 2-qubit gates. These largest-scale demonstrations for the QPE-type algorithm represent significant progress not only toward practical applications of near-term quantum computing but also toward preparation for the era of error-corrected quantum devices.