Session K51: Quantum Error Mitigation Algorithms

Recent Progress in Quantum Error Mitigation

Quantum computers, with their potential for revolutionising various computation tasks, face a significant roadblock in the form of errors due to imperfect manipulations and unwanted interactions. The implementation of full quantum error correction is still some distance away due to its requirement for high-quality qubit operations and its qubit overhead. This inspires the field of quantum error mitigation, which is applicable to the noisy intermediate-scale machines we have today and is also expected to play an important role in the early fault-tolerant era in which our qubit numbers are still limited. In this talk, I am going to give a quick introduction and overview of quantum error mitigation to people who are unfamiliar with the field. Then I am going to expand on some of the recent quantum error mitigation techniques developed in our group.   

Impact of Error Mitigation and Error Detection on Noise-induced Barren Plateau

Variational quantum algorithms (VQA) suffer from the noise-induced barren plateau (NIBP) problem during training process. In this scenario, the landscape of the cost function becomes exponentially flat due to noise. Both error mitigation (EM) and error detection codes (ED) with post-selection have demonstrated their efficacy in reducing the impact of noise on near-term quantum computers. A natural inquiry that arises is whether EM and ED can alleviate the NIBP issue. In this project, we assess the impact of various EM and ED methods on QAOA and report its approximation ratio in relation to the increase in noise level (number of layers in the circuit). We offer a theoretical analysis based on our proposed noise model and present experimental results obtained from IBM quantum hardware.   

Cascaded variational quantum eigensolver calculations of molecular and ionic compounds

We present results of many-electron ground-state calculations of molecular and ionic compounds obtained using the cascaded variational quantum eigensolver (CVQE) algorithm. We focus on molecules and ions that are involved in chemical reactions that contribute to aqueous protonic diffusion, which is one of many mechanisms involved in corrosion. This includes the hydronium cation H₃O⁺, which has 8 valence electrons occupying many-electron states constructed from 14 valence spin-orbitals.  The results include CVQE data obtained using quantum simulators as well as IonQ quantum hardware. The IonQ Aria QPU has 25 algorithmic qubits with excellent gate fidelities and all-to-all qubit connectivity, which offers a great deal of flexibility to optimize the CVQE circuits by reducing the number of two-qubit entangling gates.  Pairing the optimized circuit with error detection techniques, we execute the CVQE quantum circuits on the Aria QPU to calculate the many-electron ground-state energy of the hydronium cation.   

Error Analysis for the Estimated Energy from a [[4,2,2]] Encoded Variational Quantum Eigensolver Ansatz

Application benchmarks that run on noisy intermediate scale quantum computing (NISQ) devices require techniques for detecting and mitigating errors to assess accuracy and performance. Quantum error detection codes provide a framework in which to encode these computations and track the presence of errors, but the subsequent logical error rate depends on the application circuit as well as the underlying hardware noise. Here we extend recent results that show that the [[4,2,2]] error detection code improves the accuracy of computational chemistry calculations of an encoded variational ansatz. Within the context of the variational quantum eigensolver (VQE), we numerically simulate the mixed state generated by noisy execution of a unitary coupled-cluster doubles ansatz of the hydrogen molecule, accounting for variations in circuit parameters due to noise and the underlying noise models. Simulations of the unencoded, encoded, and post-selected states lead to estimates of energy, logical fidelity and probabilities for error-free calculations. We analyze the precision and accuracy of the estimated energy against the benchmark of chemical accuracy of 1mHa. We find that post-selection improves accuracy of the energy estimate over the unencoded ansatz and decreases precision due to the loss of samples. The estimated energy is found to be within chemical accuracy at 0.08% noise from 49.6% samples retained after post-selection from 200000 samples, and it is nearly 2 mHa lower than the corresponding unencoded simulation.   

Error Mitigation, Optimization, and Extrapolation on a Trapped Ion Testbed

Zero Noise Extrapolation (ZNE) is useful as an error mitigation technique because it is broadly applicable to a variety of quantum devices and applications. We test three implementations of ZNE on the Quantum Scientific Computing Open User Testbed (QSCOUT) ion-trap device at Sandia National Laboratories. We study ZNE in the context of the Variational Quantum Eigensolver (VQE), specifically using VQE to solve the electronic structure problem for HeH+. Our experimental results show that a naive implementation of ZNE via increasing the duration of our two-qubit gates does not scale our device noise enough for extrapolation. Similarly, scaling the two-qubit gate detuning only accounted for some of the noise present in our experiment. Instead, a gate-based, unitary folding noise scaling approach wherein two-qubit identity operations are added to the circuit proved amenable to error mitigation. Fitting a linear function to this data, we obtained noiseless energy estimates with 30x less error than an unmitigated estimate, 4 millihartree away from the ground state energy. This result shows that while ZNE can be used on almost any device, it is important to tailor noise scaling methods to the hardware. In addition, ZNE can be used as a practical benchmark for probing the most dominant sources of noise in experiment.   

Quantum computer simulation of near-surface oxygen vacancies in α-Al2O3 (0001)

We study the technologically relevant α-Al₂O₃ (0001) surface, which is known to host applications such as catalysis, and naturally occurring processes such as corrosion. Near-surface oxygen vacancies are critical to describe these processes, and in this talk we provide an in-depth analysis of the vacancy defects in α-Al₂O₃ (0001) surface with density functional theory and quantum computer simulations.

Employing first-principles calculations we describe the electronic properties of pristine α-Al₂O₃ (0001) surface and of near-surface O vacancies. Using a hybrid exchange-correlation functional (HSE06), we first obtain the relaxed pristine (0001) surface with Al termination. These geometries are consistent with X-ray diffraction results. Upon introducing an O vacancy, a highly non-dispersive in-gap electronic defect state appears that was absent for the pristine surface. Its band-decomposed charge density and that of the second unoccupied state reveal a strong charge localization near the O vacancy and the adjacent surface Al atom.

Next, we study these vacancy states with quantum defect embedding theory (QDET) calculations to unravel their ground and excited-state properties. We define an active space consisting of strongly localized states near the defect and treat the remainder as environment. An effective Hamiltonian is solved for the active space which includes the effective screening from the environment within the random phase approximation to obtain the eigenvalues with full configurational interaction (FCI).

Furthermore, we solve the effective Hamiltonian on a quantum computer by employing a model consisting of an active space of one occupied and one unoccupied band from the QDET calculation. On a four-qubit circuit with a Unitary coupled-cluster (UCC-3) ansatz, we calculate the ground-state energy for the active space with Variational quantum eigensolver (VQE). On a noiseless simulator, we achieve excellent agreement with the reference FCI values. Upon running the calculations on a quantum computer, we obtain an error of 0.28±0.04 eV owing to the noise in the hardware. To mitigate noise, we employed Zero-Noise extrapolation with global folding and successfully reduced the error to 0.09±0.04 eV.   

Demonstrating Quantum Computation for Quasiparticle Band Structures

Understanding and predicting the properties of solid-state materials from first-principles has been a great challenge for decades. Owing to the recent advances in quantum technologies, quantum computations offer a promising way to achieve this goal. Here, we demonstrate the first-principles calculation of a quasiparticle band structure on actual quantum computers. Specifically, the ground state is calculated by a quantum classical hybrid algorithm, variational quantum eigensolver, and the band structure is determined by quantum subspace expansion based on the ground state. In order to realize calculations with the current noisy intermediate-scale quantum devices, we have applied a qubit reduction method and error mitigation techniques such as readout error mitigation and zero noise extrapolation. In addition, by repeating the measurements several times at different times and taking their average, the influence of time variation of noise turned out to be mitigated and we obtained more accurate results. Our demonstration will pave the way to practical applications of quantum computers.   

Qubit-Efficient Variational Quantum Optimization for Matching Problems with Quadratic Constraints

Variational Quantum Optimization is a promising approach for solving hard constrained combinatorial optimization problems in the NISQ era since it leverages shallow circuits. However, it is difficult to encode inequality constraints in the quantum circuit because this typically requires a large number of ancillary slack qubits. In addition, it is difficult to model problem instances at industrially relevant scales due to low qubit availability in the NISQ era. In this work we introduce a qubit-efficient problem encoding and a novel cost function that aims to push the limits of solving matching problems with quadratic constraints on IonQ quantum computers. This problem has commercial applications in areas like cargo loading, scheduling, and resource allocation, amongst others. We present results obtained using quantum simulators as well as IonQ's Forte QPU, which boasts 29 algorithmic qubits with excellent gate fidelities. The QPU's all-to-all qubit connectivity offers great flexibility for optimizing the quantum circuits by reducing the number of required entangling two-qubit gates. When paired with error mitigation techniques, we obtain optimal solutions with high sampling probability upon executing quantum circuits on the Forte QPU.   

Enabling Hybrid Algorithms on Utility-Era Devices via Error Suppression

While scalable and fault-tolerant quantum computers are currently out of reach, hybrid quantum-classical algorithms provide a path toward achieving quantum advantage for certain types of optimization problems on utility-era devices. Yet, errors and imperfections in existing quantum computers degrade the performance of these algorithms, rendering them unavailing. Various statistical techniques, such as ZNE and PEC, have been used to address this challenge. However, these methods can be applied to only a small subset of problems, and introduce an extensive sampling overhead, increasing the time and cost required to complete these tasks. We show that a deterministic error suppression workflow, with no overhead, improves the performance of hybrid algorithms on currently available quantum hardware by addressing both the quality of implementation of individual circuits and the effectiveness of the classical optimization loop. In this talk, we review the different parts of our workflow and demonstrate its effectiveness via the implementation of QAOA on real devices. We show orders of magnitude improvement in performance alongside a substantial reduction in the resources needed for achieving these tasks.

In particular, we show that we can consistently, with over 99% success rate, correctly solve combinatorial optimization problems, such as MaxCut and Min Vertex Cover, using minimal resources on >40Q superconducting quantum devices.   

Quantifying the Role of Correlated Noise in the Quantum Approximate Optimization Algorithm

The quantum approximate optimization algorithm (QAOA) is a promising application for solving optimization problems on noisy intermediate-scale quantum devices. However, their potential for quantum advantage is hindered by the presence of noise stemming from unwanted interactions between quantum systems and their environments. In prominent quantum technologies, like superconducting qubits, relevant noise sources have been found to possess both spatial and temporal correlations. While previous studies have sought to investigate the impact of noise and potential avenues for robustness in QAOA in the presence of Markovian noise, little is known about the effect of spatiotemporal correlated noise on QAOA performance. In this study, we investigate the impact of correlated noise on the QAOA variant of Grover's search using the filter function formalism. We assess the robustness of QAOA by optimizing the filter function via the variational parameters. We show that analytical error bounds relating the noise statistical properties to the approximation ratio and training error can be derived.   

Scaling from utility to advantage for the transverse-field Ising model with software

Recent evidence for utility on noisy quantum computers has ushered a new era of quantum computing. Given the astonishing progress of quantum hardware, existing quantum software need to accommodate for these advancements by showing that they can scale from the era of utility to the era of error correction. The application that provides evidence for utility, the 2D Transverse-Field ising model (TFIM) happens to also be one of the earliest applications that will achieve practical quantum advantage in the era of error corrected quantum computers. Using techniques like Dynamical decoupling, Zero-Noise extrapolation, optimized decomposition, Echoed Cross-Resonance, Hidden Inverses, Equivalent Circuit Averaging and Stabilizer slicing we show how quantum software can scale by running physical TFIM on noisy quantum computers, turning the noisy quantum computers into error corrected quantum computers and running logical versions of TFIM on the same devices.   

A critical limitation of quantum imaginary time evolution-like algorithms in the noisy quantum hardware

The variational quantum imaginary time evolution algorithm is efficient in finding the ground state of a quantum Hamiltonian. This algorithm involves solving a system of linear equations in a classical computer and the solution is then used to propagate a quantum wavefunction. Here, we show that owing to the noisy nature of current quantum processors, such a quantum algorithm or the family of quantum algorithms that require classical computation of inverting a matrix with high condition number will require single- and two-qubit gates with very low error probability. Failure to meet such conditions will result in erroneous quantum data propagation even for a relatively small quantum circuit ansatz. Specifically, we find the upper bounds on how the quantum algorithmic error scales with the probability of errors in quantum hardware. Our work challenges the mainstream notion of hybrid quantum-classical quantum algorithms being able to perform under noisy environments while we show such algorithms in fact require very low error quantum gates to get reliable results.   

Quantum Speedup on Limited Hamming Weight Simon's Problem

Many quantum algorithms have been theoretically shown to outperform their classical counterparts in solving problems of increasing size. However, in today's noisy intermediate-scale quantum (NISQ) devices, it is still difficult to demonstrate practical speedup in quantum computers (QCs) due to noise that leads to computational errors. Here, we demonstrate an algorithmic speedup for Simon's problem for oracles with Hamming weight (HW) up to 8 using the number-of-oracle-queries-to-solution (NTS) metric, which scales with problem size. The experiments were performed on two different 127-qubit IBM Quantum superconducting processors. The speedup on HW up to 7 is observed on ibm\_sherbrooke only when the computation is protected by dynamical decoupling (DD) and measurement error mitigation (MEM) is used to simulate the absence of measurement errors.