Quantum phase estimation (QPE) serves as a building block of many different quantum algorithms and finds important applications in computational chemistry problems. Despite the rapid development of quantum hardware, experimental demonstration of QPE for chemistry problems remains challenging due to its large circuit depth and the lack of quantum resources to protect the hardware from noise with fully fault-tolerant protocols. In the present work, we take a step towards fault-tolerant quantum computing by demonstrating a QPE algorithm on a Quantinuum trapped-ion computer. We employ a Bayesian approach to QPE and introduce a routine for optimal parameter selection, which we combine with a [[ n + 2 , n , 2 ]] quantum error detection code carefully tailored to the hardware capabilities. As a simple quantum chemistry example, we take a hydrogen molecule represented by a two-qubit Hamiltonian and estimate its ground state energy using our QPE protocol. In the experiment, we use the quantum circuits containing as many as 920 physical two-qubit gates to estimate the ground state energy within 6 × 10 − 3 hartree of the exact value. Published by the American Physical Society 2024
Quantum phase estimation (QPE) serves as a building block of many different quantum algorithms and finds important applications in computational chemistry problems. Despite the rapid development of quantum hardware, experimental demonstration of QPE for chemistry problems remains challenging due to its large circuit depth and the lack of quantum resources to protect the hardware from noise with fully fault-tolerant protocols. In the present work, we take a step towards fault-tolerant quantum computing by demonstrating a QPE algorithm on a Quantinuum trapped-ion computer. We employ a Bayesian approach to QPE and introduce a routine for optimal parameter selection, which we combine with a [[ n + 2 , n , 2 ]] quantum error detection code carefully tailored to the hardware capabilities. As a simple quantum chemistry example, we take a hydrogen molecule represented by a two-qubit Hamiltonian and estimate its ground state energy using our QPE protocol. In the experiment, we use the quantum circuits containing as many as 920 physical two-qubit gates to estimate the ground state energy within 6 × 10 − 3 hartree of the exact value. Published by the American Physical Society 2024
A number of exciting recent results have been seen in the field of quantum error correction. These include initial demonstrations of error correction on current quantum hardware and resource estimates which improve understanding of the requirements to run large-scale quantum algorithms for real-world applications. In this work, we bridge the gap between these two developments by performing careful estimation of the resources required to fault-tolerantly perform quantum phase estimation (QPE) on a minimal chemical example. Specifically, we describe a detailed compilation of the QPE circuit to lattice surgery operations for the rotated surface code, for a hydrogen molecule in a minimal basis set. We describe a number of optimizations at both the algorithmic and error correction levels. We find that implementing even a simple chemistry circuit requires 1000 qubits and 2300 quantum error correction rounds, emphasising the need for improved error correction techniques specifically targeting the early fault-tolerant regime. Published by the American Physical Society 2024
We present the first demonstration of an end-to-end pipeline with quantum error correction (QEC) for a quantum computation of the electronic structure of molecular systems. We calculate the ground-state energy of molecular hydrogen, using quantum phase estimation (QPE) on qubits encoded with the [[7,1,3]] color code on Quantinuum H2--2. We obtain improvements in computational fidelity by (1) introducing several partially fault-tolerant (FT) techniques for the Clifford+RZ (arbitrary-angle single-qubit rotation) gate set, and (2) integrating Steane QEC gadgets for real-time error correction. In particular, the latter enhances the QPE circuits' performance despite the complexity of the extra QEC circuitry. The encoded circuits contain up to 1585 (546) fixed and 7202 (1702) conditional physical two-qubit gates (mid-circuit measurements), and $\sim3900(\sim$760) total operations are applied on average. The energy E is experimentally estimated to within E−EFCI=0.001(13) hartree, where EFCI denotes the exact ground state energy within the given basis set. Additionally, we conduct numerical simulations with tunable noise parameters to identify the dominant sources of noise. We find that orienting the QEC protocols towards higher memory noise protection is the most promising avenue to improve our experimental results.
Quantum-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary N tot times, the variance of the estimates can scale as O ( 1 / N tot 2 ) , compared to the best “classical” strategy with O ( 1 / N tot ) . The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without interprobe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling O ( 1 / N tot 2 ) . Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analizes the results of previous steps. Published by the American Physical Society 2024
Emergent quantum computing technologies are widely expected to provide novel approaches in the simulation of quantum chemistry. Despite rapid improvements in the scale and fidelity of quantum computers, high resource requirements make the execution of quantum chemistry experiments challenging. Typical experiments are limited in the number of qubits used, incur a substantial shot cost, or require complex architecture-specific optimization and error mitigation techniques. In this paper, we propose a conceptually simple benchmarking approach involving the use of multi-ancilla quantum phase estimation. Our approach is restricted to very small chemical systems, and does not scale favorably beyond molecular systems that can be described with 2 qubits; however, this restriction allows us to generate circuits that scale quadratically in gate count with the number of qubits in the readout register. This enables the execution of quantum chemistry circuits that act on many qubits, while producing meaningful results with limited shot counts. We use this technique (with 200 shots per experiment) to calculate the ground state energy of molecular hydrogen to 50 bits of precision (8.9×10−16 hartree) on a 56-qubit trapped-ion quantum computer, negating Trotter error. Including Trotter error, we obtain between 32 and 36 bits of precision (1.5∗10−10 and 6.0∗10−11 hartree respectively), vastly exceeding chemical accuracy (1.6∗10−3 hartree) against Full Configuration Interaction. We consider application of the approach to deeper circuits, and discuss potential as a benchmark task for near-term quantum devices.
Quantum simulation of molecular electronic structure is one of the most promising applications of quantum computing. However, achieving chemically accurate predictions for strongly correlated systems requires quantum phase estimation (QPE) on fault-tolerant quantum computing (FTQC) devices. Existing resource estimates for typical FTQC architectures suggest that such calculations demand millions of physical qubits, thereby placing them beyond the reach of near-term devices. Here, we investigate the feasibility of performing QPE for chemically relevant molecular systems in an early-FTQC regime, characterized by partial fault tolerance, constrained qubit budgets, and limited circuit depth. Our framework is based on single-ancilla, Trotter-based QPE implementations combined with partially randomized time evolution. Within this framework, we develop a novel Hamiltonian optimization strategy, termed unitary weight concentration, that reduces algorithmic cost by reshaping linear-combination-of-unitaries representations. Applying this framework to active-space models of iron-sulfur clusters, cytochrome P450 active sites, and CO2-utilization catalysts, we perform end-to-end resource estimation using the latest version of the space-time efficient analog rotation (STAR) architecture. Our results indicate that ground-state energy estimation for active spaces of approximately 20 to 50 spatial orbitals, well beyond the reach of classical full configuration interaction, is achievable using ∼105 physical qubits, with runtimes on the order of days to weeks. These findings demonstrate that while full-fledged fault-tolerant quantum computers will be required for even larger molecular simulations, chemically meaningful quantum chemistry problems are already within reach in an experimentally relevant, early-FTQC regime.
We compare several quantum phase estimation (QPE) protocols intended for early fault-tolerant quantum computers (EFTQCs) in the context of models of their implementations on a surface code architecture. We estimate the logical and physical resources required to use these protocols to calculate the ground state energy of molecular hydrogen in a minimal basis with error below 10−3 atomic units in the presence of depolarizing logical errors. Accounting for the overhead of rotation synthesis and magic state distillation, we find that the total T-gate counts do not vary significantly among the EFT QPE protocols at fixed state overlap. In addition to reducing the number of ancilla qubits and circuit depth, the noise robustness of the EFT protocols can be leveraged to reduce resource requirements below those of textbook QPE, realizing approximately a 300-fold reduction in computational volume in some cases. Even so, our estimates are well beyond the scale of existing early fault-tolerance demonstrations.