Important quantum algorithm routines allow the implementation of specific quantum operations (a.k.a. gates) by combining basic quantum circuits with an iterative structure. In this structure, the number of repetitions of the basic circuit pattern is associated to convergence parameters. This iterative structure behaves similarly to function approximation by series expansion: the higher the truncation order, the better the target gate (i.e. operation) approximation. The asymptotic convergence of the gate error with respect to the number of basic pattern repetitions is known. It is referred to as the query complexity. The underlying convergence law is bounded, but not in an explicit fashion. Upper bounds are generally too pessimistic to be useful in practice. The actual convergence law contains constants that depend on the joint properties of the matrix encoded by the query and the initial state vector, which are difficult to compute classically. This paper proposes a strategy to study this convergence law and extract the associated constants from the gate (operation) approximation at different accuracy (convergence parameter) constructed directly on a Quantum Processing Unit (QPU). This protocol is called Numerical Error Extraction by Quantum Measurement Algorithm (NEEQMA). NEEQMA concepts are tested on specific instances of Quantum Signal Processing (QSP) and Hamiltonian Simulation by Trotterization. Knowing theexact convergence constants allows for selecting the smallest convergence parameters that enable reaching the required gate approximation accuracy, hence satisfying the quantum algorithm's requirements.
Important quantum algorithm routines allow the implementation of specific quantum operations (a.k.a. gates) by combining basic quantum circuits with an iterative structure. In this structure, the number of repetitions of the basic circuit pattern is associated to convergence parameters. This iterative structure behaves similarly to function approximation by series expansion: the higher the truncation order, the better the target gate (i.e. operation) approximation. The asymptotic convergence of the gate error with respect to the number of basic pattern repetitions is known. It is referred to as the query complexity. The underlying convergence law is bounded, but not in an explicit fashion. Upper bounds are generally too pessimistic to be useful in practice. The actual convergence law contains constants that depend on the joint properties of the matrix encoded by the query and the initial state vector, which are difficult to compute classically. This paper proposes a strategy to study this convergence law and extract the associated constants from the gate (operation) approximation at different accuracy (convergence parameter) constructed directly on a Quantum Processing Unit (QPU). This protocol is called Numerical Error Extraction by Quantum Measurement Algorithm (NEEQMA). NEEQMA concepts are tested on specific instances of Quantum Signal Processing (QSP) and Hamiltonian Simulation by Trotterization. Knowing theexact convergence constants allows for selecting the smallest convergence parameters that enable reaching the required gate approximation accuracy, hence satisfying the quantum algorithm's requirements.
Rydberg atom arrays have recently emerged as one of the most promising platforms for quantum simulation and quantum information processing. However, as is the case for other experimental platforms, the longer-term success of the Rydberg atom arrays in implementing quantum algorithms depends crucially on their robustness to gate-induced errors. Here we show that, for an idealized biased-error model based on Rydberg atom dynamics, the implementation of quantum signal processing (QSP) protocols can be made error robust, in the sense that the asymptotic scaling of the gate-induced error probability is slower than that of gate complexity. Moreover, our numerical results that use experimentally accessible parameters indicate that QSP iterates made out of more than 100 gates can be implemented with constant error probability. To showcase our approach, we provide a concrete blueprint to implement QSP-based near-optimal Hamiltonian simulation on the Rydberg atom platform. The proposed protocol substantially improves both the scaling and the overhead of gate-induced errors in comparison to those protocols that implement a fourth-order product formula. Published by the American Physical Society 2024
To process a problem with a Quantum Processing Unit (QPU), it must be transformed into a sequence of quantum operators, or gates. These operators are either packed into a query (i.e. quantum algorithm primitive) that encodes the problem, or used to construct the cost function for Variationnal Quantum Algorithm (VQA). Typical queries are the problem Hamiltonian Simulation (HS) and the problem Block-Encoding (BEing). To construct the circuits associated with the quantum description, the problem must be mapped as a Linear Combination of Hamiltonian (LCH) or a Linear Combination of nitaries (LCU) matrices. All the summed Hamiltonian matrices or unitary matrices must have a known decomposition in basic gates. The complexity of this query should be incorporated into the quantum algorithm's query complexity, thereby limiting the processing possibilities of QPU for many problems. In this work, we propose Hamiltonian matrices used to map the problem of interest, and that behave like a single qubit when expressed in the appropriate basis. It leads to a framework, the Special Tripotent Hamiltonian (STH) Framework, able to implement most of the typical problems considered for quantum computing. These methods address many problems implemented on QPUs, ranging from second-quantization chemistry operators to graphs associated with Partial Differential Equations (PDE), sparse matrices, and higher-order optimization problems. This work underlines interesting properties associated with the STH basic gate decomposition. These include the ability to switch between LCH and LCU, map non-Hermitian problems, and construct the quantum circuit queries required for quantum computing. We also provide a list of STH that are used for the matrix decomposition of many structured matrices. These structured matrices are associated with graph adjacency matrices that can be combined to implement structured matrices.
An important issue in the implementation of a quantum computer is to protect quantum information from decoherence. In fault-tolerant quantum computation, the circuits used to measure the error syndromes are themselves faulty; to minimize the effect of syndrome measurement errors, the syndromes are measured repeatedly. This paper introduces a scheme based on classical codes to make this process more robust and/or reduce the needed resources and measurement time. We analyze particular implementations based on low-density generator matrix (LDGM) codes using EXIT functions.
Quantum simulation of lattice gauge theories offers a path to first-principles calculations of real-time dynamics and finite-density phenomena that are inaccessible to classical Euclidean Monte Carlo methods. Realizing this potential requires a pipeline of controlled approximations---lattice discretization, truncation, digitization, state preparation, time evolution, measurement, and continuum extrapolation---each contributing errors and computational costs. General algorithmic asymptotic complexity arguments capture only part of the cost picture: the practical resources required for end-to-end simulation depend on numerical pre-factors, the cost of the subroutines that various algorithms call, and structural features of the lattice theory, all of which the asymptotic scaling leaves unspecified. This thesis develops contributions that aim to make such dependencies explicit and quantitative. We begin with a rigorous analysis of the asymptotic gate complexity of product formulas and Quantum Signal Processing (QSP) for the time evolution of general classes of lattice Hamiltonians, identifying the parameter regimes in which each method is favorable and estimating the associated pre-factors numerically. Since the cost of QSP-based simulation is dominated (in regimes of practical interest) by the block-encoding subroutine it calls, we also develop several novel block-encoding constructions tailored to bosonic lattice field theories and show that signal-processing techniques can be repurposed as highly efficient block-encoding subroutines. We then turn to the problem of truncating the Kogut--Susskind electric-field Hilbert space. Existing error bounds rely on energy-conservation arguments and require the truncation to grow with simulation time. By exploiting Hilbert space fragmentation we derive the tightest error bounds known to date: they decay factorially in the truncation parameter and are independent of simulation time. We validate them numerically for U(1) pure gauge theory and the Schwinger model. Finally, we address the continuum limit in the presence of approximate time evolution. For product formulas, we show that Trotter errors correspond to irrelevant operators that vanish in the continuum limit, which yields a renormalization trajectory independent of the Trotter step size. More generally, we introduce the Statistically-Bounded Time Evolution (SBTE) protocol, an algorithm-independent framework that treats approximation errors as a systematic uncertainty to be driven below the working statistical precision. This decouples the continuum-limit prescription from the choice of time-evolution algorithm and provides a basis for end-to-end cost comparisons across simulation methods.
The implementation and practicality of quantum algorithms hinge largely on the quality of operations within a quantum processor. Therefore, including realistic error models in quantum computing simulation platforms is crucial for testing these algorithms. Existing classical simulation techniques of quantum information processing devices exhibit a tradeoff between scalability (the number of qubits that can be simulated) and accuracy (how close the simulation is to the target error model). In this paper, we introduce a simulation approach that relies on approximating the density matrix evolution with a stochastic sum of unitary and measurement channels within a pure-state simulation environment. This model shows an improvement of at least one order of magnitude in terms of accuracy compared to the best known stochastic approaches while allowing us to simulate a larger number of qubits compared to the exact density matrix simulation. Furthermore, we used this approach to realistically simulate Grover's algorithm and the surface code 17 using a gate set tomography characterization of quantum operations as a noise model.
Quantum computing has emerged as a transformative approach for tackling complex problems in quantum chemistry, particularly in simulating multielectron systems and electron-phonon interactions. However, the current noisy intermediate-scale quantum (NISQ) devices face significant challenges in implementing practical quantum algorithms due to error accumulation caused by increased circuit depth, qubit counts, and gate operations. To address these challenges, we present TenCirChem, an open-source Python package specifically designed for implementing variational quantum algorithms in quantum computational chemistry. TenCirChem demonstrates exceptional performance in simulating unitary coupled-cluster circuits through its innovative use of compact representations for quantum states and excitation operators. This package supports noisy circuit simulation and provides advanced algorithms for variational quantum dynamics, enabling researchers to explore complex chemical phenomena. Its capabilities are exemplified in various applications, including the calculation of potential energy curves and the investigation of quantum gate error impacts on molecular systems. Moreover, TenCirChem’s seamless integration with real quantum hardware makes it a versatile tool for both simulation and experimentation. A key innovation developed within the TenCirChem framework is the Clifford-based Hamiltonian engineering approach for molecules (CHEM). This algorithm addresses the critical challenge of achieving chemical accuracy with shallow quantum circuits, a fundamental requirement for practical applications on NISQ devices. CHEM employs a sophisticated Clifford-based Hamiltonian transformation that operates within the variational quantum eigensolver (VQE) framework using hardware-efficient ansatz. The method ensures four crucial advantages: (1) generation of initial circuit parameters corresponding to Hartree-Fock energy, (2) maximization of initial energy gradients with respect to circuit parameters, (3) minimal classical processing overhead without additional quantum resource requirements, and (4) compatibility with any circuit topology. Through quantum hardware emulator demonstrations, CHEM has achieved chemical accuracy for systems up to 12 qubits with fewer than 30 two-qubit gates, representing a significant advancement in practical quantum computational chemistry. To enhance the efficiency of variational quantum algorithms, we developed the sequential optimization with approximate parabola (SOAP) method, specifically designed for parameter optimization in unitary coupled-cluster ansatz. SOAP addresses the critical bottleneck of measurement requirements in VQE by implementing an innovative optimization strategy that approximates the energy landscape as quadratic functions. This approach minimizes the number of energy evaluations while incorporating parameter correlations through the integration of average directions from previous iterations. Benchmark studies demonstrate SOAP’s superior performance, showing faster convergence and enhanced noise robustness compared to traditional optimization methods. The method’s scalability has been validated through numerical simulations of up to 20 qubits, and its practical efficacy has been confirmed through experiments on superconducting quantum computers. For simulating electron-phonon systems, we introduce a variational basis state encoding algorithm that significantly reduces resource requirements compared to conventional unary and binary encoding schemes. Our approach achieves smaller scaling than traditional methods for qubits and gates for systems obeying the area law of entanglement entropy, this remarkable reduction in resource requirements comes at the cost of a constant amount of additional measurements. The algorithm’s effectiveness has been validated through both numerical simulations and quantum hardware experiments, demonstrating that using just one or two qubits per phonon mode can produce quantitatively accurate results across various coupling regimes. The integration of these innovations within the TenCirChem library represents a significant advancement in quantum computational chemistry. The software package provides researchers with a comprehensive toolkit for developing, testing, and implementing quantum algorithms, while the novel methods address fundamental challenges in circuit depth, parameter optimization, and resource efficiency. These developments collectively enhance the practicality of quantum computing for addressing real-world chemical problems in the NISQ era, offering improved accuracy, efficiency, and noise robustness.