Session M51: Variational Quantum Algorithms on Noisy Hardware

Good and Bad News for Noisy Variational Quantum Algorithms – Part I

Variational quantum algorithms (VQAs) have been regarded as the most promising route towards useful, near-term quantum advantage. The hope has been that the variational quantum eigensolver and the quantum approximate optimisation algorithm could solve classically intractable problems in chemistry and binary classification, respectively. Recent years have seen the development of shallow quantum circuits for these purposes. In this talk, we present a thorough analysis of how noise affect these algorithms' performances. We show that for leading algorithms for variational quantum chemistry, low levels of gate noise preclude useful quantum simulations of even the smallest molecules. For VQAs to output chemically-accurate eigen-energy estimates, noise levels must be significantly lower than fault-tolerance thresholds. The same unfortunate conclusions hold for VQAs targeted towards binary classification. We evaluate possible remedies. We show that simple error-mitigation methods can boost the noise resilience by a factor of 10 to 100. Further, we provide additional tricks to boost noise resilience. Using knowledge of the algorithmic properties of VQAs, we show that it is possible to dynamically tweak algorithms to avoid computational regions that incur significant noise.   

Good and Bad News for Noisy Variational Quantum Algorithms – Part II

The quantum approximate optimization algorithm (QAOA) is an appealing proposal to solve NP problems on noisy intermediate-scale quantum (NISQ) hardware. Making NISQ implementations of the QAOA resilient to noise requires short ansatz circuits with as few CNOT gates as possible. In this talk, we present Dynamic-ADAPT-QAOA. Our algorithm significantly reduces the circuit depth and the CNOT count of standard ADAPT-QAOA, a leading proposal for near-term implementations of the QAOA. Throughout our algorithm, the decision to apply CNOT-intensive operations is made dynamically, based on algorithmic benefits. Using density-matrix simulations, we benchmark the noise resilience of ADAPT-QAOA and Dynamic-ADAPT-QAOA. We compute the gate-error probability p below which these algorithms provide, on average, more accurate solutions than the classical, polynomial-time approximation algorithm by Goemans and Williamson. For small systems with 6−10 qubits, we show that p > 0.001 for Dynamic-ADAPT-QAOA. Compared to standard ADAPT-QAOA, this constitutes an order-of-magnitude improvement in noise resilience. This improvement should make Dynamic-ADAPT-QAOA viable for implementations on superconducting NISQ hardware, even in the absence of error mitigation.   

On Penalty Functions for Variational Quantum Algorithms

Situated between a dedicated quantum emulator—which bootstraps one quantum system to mimic another—and a fully programmable quantum processor, lies the celebrated gate-based quantum processors of today. For utilitarian purposes, today's quantum processors embody what is known as the variational or hybrid-quantum-classical model of quantum computation. While theoretically universal for quantum computation, the practical appeal of the variational model lies in maximizing the use-case capacity of low-depth, parameterized quantum circuits, albeit at the cost of outer loop classical optimization. This model, a Hamiltonian-based computation framework, hinges on minimizing an energy function to prepare an approximate ground state. This talk presents recent findings related to penalty function constructions.   

Impact of incoherent noise on Grover and quantum phase estimation algorithms

The Grover search and phase estimation are two fundamental quantum algorithms. The major challenge in running these two and other quantum algorithms is the noise in quantum computers. This noise is due to the interactions of qubits with the environment and faulty gate operations. Here, we present the impact of incoherent noise on two algorithms. The noise impact is modeled as trace-preserving and completely positive quantum channels. Different noise models such as depolarizing, phase flip, bit flip, and bit-phase flip are taken to understand the performance of these algorithms in the presence of noise. The simulation results indicate that the probability of success of the Grover algorithm and the standard deviation of the eigenvalue of the unitary operator have strong exponential dependence upon the error probability of individual qubits. Furthermore, the original formulation is compared with the recently proposed generalization in terms of singular value transformation.   

Noise-induced transition in optimal solutions of variational quantum algorithms

Variational quantum algorithms are widely considered a promising class of near-team algorithms that may demonstrate a practical quantum advantage in problems of scientific interest. One of the major obstacles to a scalable realization is the difficulty in optimizing the noisy cost function. In this work, we investigate the effect of noise on the optimization. By studying a simple model, we observe an abrupt transition induced by noise to the optimal solution. We will present the numerical simulation, experimental demonstration using IBM QPUs, and theoretical analysis indicating that similar transitions exist beyond the simple model. Our results suggest that a careful examination of the optimal solutions is necessary to ensure the qualitative correctness of the solutions when implementing variational algorithms on NISQ hardware.   

Engineered dissipation to mitigate barren plateaus

Variational quantum algorithms provide a promising solution to optimization problems on noisy quantum computers, with applications ranging from chemistry to machine learning. These algorithms rely on efficient quantum circuit training, which can be affected by problems such as barren plateaus. Our research has uncovered a new method, based on dissipation, to mitigate this problem. Despite the negative effects of dissipation and noise on quantum algorithms, we show that the systematic inclusion of engineered Markovian losses following each unitary quantum circuit layer enables efficient trainability of quantum variational models. We define the precise properties that these dissipative methods must exhibit and establish their efficient optimization. In addition to analytically proving the absence of barren plateaus, we have also numerically tested our theory in both a synthetic and a quantum chemistry example. The results indicate that our strategy can make substantial progress in various domains, highlighting its versatility.   

Approximate Quantum Algorithms for Efficient Classical-to-Quantum Data Loading and Encoding

Classical information loading is a crucial task for many quantum algorithms, playing a fundamental role in the field of quantum machine learning. Consequently, the inefficiency of this loading process becomes a significant bottleneck for the application of these algorithms. In this context, we present and compare algorithms for the amplitude and dynamic encodings of classical data into a quantum computer. 

In the case of amplitude encoding, we introduce two approximate quantum-state preparation methods for the NISQ era, drawing inspiration from the Grover-Rudolph algorithm. The first method reduces the number of gates required when no ancillary qubits are used, while the second proposes a variational algorithm capable of loading real functions beyond the Grover-Rudolph algorithm. We also examine the encoding of polynomial functions, either through their matrix product state representation or a scheme that involves the block encoding of the linear function using the Walsh-Hadamard transform and a polynomial transformation of the amplitudes, achieved through the quantum singular value transformation (QSVT). 

On the other hand, in the context of dynamic encoding, we enhance the matrix exponentiation techniques when a limited number of copies of a quantum state is available. This is achieved by introducing imperfect quantum copies, which significantly improve the performance of previous proposals.

The Authors acknowledge support from EU FET Open project EPIQUS (899368) and HORIZON-CL4- 2022-QUANTUM01-SGA project 101113946 OpenSuperQPlus100 of the EU Flagship on Quantum Technologies, the Spanish Ramón y Cajal Grant RYC-2020-030503-I, project Grant No. PID2021-125823NA-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” and “ERDF Invest in your Future”, and from the IKUR Strategy under the collaboration agreement between Ikerbasque Foundation and BCAM on behalf of the Department of Education of the Basque Government. This project has also received support from the Spanish Ministry of Economic Affairs and Digital Transformation through the QUANTUM ENIA project call - Quantum Spain, and by the EU through the Recovery, Transformation and Resilience Plan - NextGenerationEU.   

Shallow-Depth Variational Quantum Hypothesis Testing

The task of discriminating between two known quantum channels is a well known binary hypothesis testing task. We present a variational quantum algorithm with a parameterized state preparation and two-outcome positive operator valued measure (POVM) which defines the acceptance criteria for the hypothesis test. Both the state preparation and measurement are simultaneously optimized using success probability of single-shot discrimination as an objective function which can be calculated using localized measurements. Under constrained signal mode photon number quantum illumination we match the performance of known optimal 2-mode probes by simulating a bosonic circuit. Our results show that variational algorithms can prepare optimal states for binary hypothesis testing with resource constraints.   

Algorithmic Developments for Calculating Excited States on Quantum Hardware

Calculating ground state energies has been the focus of many quantum algorithms due to their importance in chemistry, physics, and material science applications. Near-term strategies include quantum-classical hybrid algorithms such as variational quantum eigensolver. Excited states are generally harder to prepare than ground states with classical algorithms and evidence suggests that the same holds true for quantum algorithms. There have been various methods to prepare the excited states in quantum computers such as variance minimization or subspace diagonalization methods.  Here we build off recent classical innovations in orbital optimization to propose a new method for calculating excited states on quantum hardware.   

Evidence of Scaling Advantage for the Quantum Approximate Optimization Algorithm on a Classically Intractable Problem

The quantum approximate optimization algorithm (QAOA) is a leading candidate algorithm for solving optimization problems on quantum computers. However, the potential of QAOA to tackle classically intractable problems remains unclear. In this paper, we perform an extensive numerical investigation of QAOA on the Low Autocorrelation Binary Sequences (LABS) problem. The rapid growth of the problem's complexity with the number of spins N makes it classically intractable even for moderately sized instances, with the best-known heuristics observed to fail to find a good solution for problems with N⪆200. We perform noiseless simulations with up to 40 qubits and observe that out to this system size, the runtime of QAOA with fixed parameters and a constant number of layers scales better than branch-and-bound solvers, which are the state-of-the-art exact solvers for LABS. The combination of QAOA with quantum minimum-finding on an idealized quantum computer gives the best empirical scaling of any algorithm for the LABS problem. We demonstrate experimental progress in compiling and executing QAOA for the LABS problem using an algorithm-specific error detection scheme on Quantinuum trapped-ion processors. Our results provide evidence for the utility of QAOA as an algorithmic component when executed on an idealized quantum computer.   

Calculating Energies in the Contextual Subspace

The contextual subspace variational quantum eigensolver (CS-VQE) is a hybrid quantum-classical algorithm that approximates the ground-state energy of a given qubit Hamiltonian. It achieves this by separating the Hamiltonian into contextual and noncontextual parts. The ground-state energy is approximated by classically solving the noncontextual problem, followed by solving the contextual problem using VQE, constrained by the noncontextual solution. In general, computation of the contextual correction needs fewer qubits and measurements compared with solving the full Hamiltonian via traditional VQE. We showcase the CS-VQE algorithm and present different improvements to the method.