Session Z51: Parameterized Quantum Circuits

Power and Limitations of Parameterized Quantum Circuits

In recent years, the study of parameterized quantum circuits (PQCs) has gained significant attention in quantum machine learning and near-term quantum information processing. These circuits are known for their potential to approximate complex functions and their ability to encode data efficiently. However, our understanding of the power and limitations of PQCs is still quite limited. In this talk, we will delve into the power and limits of parameterized quantum circuits in several essential tasks. We will analyze the expressivity and capability of PQCs in terms of approximating functions, learning quantum states, and encoding classical data. We will also discuss the potential of designing quantum algorithms by combining PQCs and existing algorithmic tools. Overall, this talk aims to shed light on the power of PQCs and their potential advantages in near-term quantum applications. This talk is mainly based on arXiv:2205.07848, 2206.08273, 2310.07528.   

Iterative optimization of hard spin glass problems with high frequency AC drives (Part I)

Performance improvements of quantum algorithms such as quantum annealing, adiabatic quantum computing and the quantum approximate optimization algorithm over smart classical algorithms are often marginal, particularly when concerns such as the overhead of computing quantum expectation values for variational circuits, and the immense prefactor disadvantages quantum hardware exhibits compared to parallel silicon, are taken into account. In light of these challenges, we introduce a new quantum algorithm called IST-SAT (Iterative-Symphonic Tunneling-Boolean Satisfiability Problem) which circumvents computing gradients and parameter optimization. IST-SAT is a novel heuristic update to high-depth QAOA that simulates evolution under high-frequency drives tuned to the problem graph, and iteratively updates the applied Hamiltonian using bitstrings from previous shots. In this first talk, we introduce the IST-SAT algorithm and sketch its mechanism and workflow.   

Iterative optimization of hard spin glass problems with high frequency AC drives (Part II)

In this talk, we benchmark the IST-SAT algorithm using sets of hard MAX3-XORSAT instances, which are exponentially difficult for both exact and approximate optimization for all known classical and quantum methods. Using the Fujitsu Quantum Simulator–a classical HPC system–we estimate the average time to solution for IST-SAT through large scale numerical simulations of up to N = 30 qubit problems. We find that when IST-SAT is "seeded" using high-depth QAOA to set initial parameters, it demonstrates significant polynomial speedups over both QAOA (which performs relatively poorly for these problems) and the best reported classical algorithms. More sophisticated initial seeding algorithms can produce even larger gains. The mechanism for IST-SAT is fairly generic and we expect it should generalize to other quantum optimization problems which are bottlenecked by exponentially small gaps and first-order transitions.   

Resource analysis of quantum algorithms for coarse-grained protein folding models

Protein folding processes are a complex and vital aspect of molecular biology that quantum devices may help model.

Here, we analyze the resource requirements for simulating protein folding on a quantum computer, assessing this problem's feasibility in the current and near-future technological landscape. We calculate the minimum number of qubits, interactions, and two-qubit gates necessary to represent the energy of a specific folding.

We study various classical folding models and assess their compatibility with diverse bit-encodings, considering the constraints of existing quantum hardware. Specifically, we focus on the resources needed to encode the Hamiltonian representing the energy function of each protein folding model concerning the chain's amino acid count. We conclude that although the number of qubits required falls within the realm of current technological capabilities, the high number of terms in the Hamiltonian, resulting in a substantial requirement for numerous quantum gates, emerges as a significant limitation.    

Variational Quantum Algorithm for Partial Singular Value Decomposition

Quantum entanglement is a key description of multi-particle correlations in various quantum systems and constitutes the fundamental basis for emerging technologies such as quantum computation and communication. Specifically, the entanglement spectrum, quantifying quantum entanglement, serves as a crucial criterion for identifying topological quantum phases and assessing quantum criticality around phase transitions. Utilizing quantum computing for Singular Value Decomposition (SVD) plays a pivotal role in understanding quantum entanglement in large-scale quantum systems beyond the capability of classical computers.

The Variational Quantum Algorithm (VQA), executable with reduced gate operations, emerges as a promising solution for current Noisy Intermediate-Scale Quantum (NISQ) devices, where the error accumulation due to gate operations poses challenges. Numerous researchers have proposed VQAs for SVD, centered on approximating the unitary matrix in SVD using quantum circuits. However, applying such quantum algorithms for highly entangled states is challenging, demanding an exponentially large number of gate operations.

Our study revisited this challenge, redefining VQA as an approximation problem for a specific quantum circuit. We discovered that our partial SVD algorithm, approximating quantum circuits corresponding to individual left-right singular vectors, accurately determines singular values with fewer gate operations than conventional quantum SVD algorithms.   

Reconfigurable Quantum Hutchinson with Real-Time Evolution

Recently, quantum algorithms leveraging dynamical evolution under the many-body Hamiltonian of interest have proven exceptionally effective in pinpointing individual eigenvalues near the edge of Hamiltonian spectrum, for example the ground state energy. This work delves into the potential of real-time evolution to evaluate the aggregate of eigenvalues across the entire spectrum. In particular, we introduce a simple near-term algorithm designed to compute the trace of a wide class of operators, including functions of the target Hamiltonian. Using stochastic real-time evolution, we transform the task of trace estimations to straightforward state-vector simulations on quantum computer. For numerical illustration, we highlight important applications, such as density of states and free energy calculations, relevant in physical, chemical, and materials sciences.   

Variational secure cloud quantum computing

Variational quantum algorithms (VQAs) have been considered useful applications of noisy intermediate-scale quantum (NISQ) devices. Typically, in VQAs, a parametrized ansatz circuit generates a trial wave function, and the parameters are optimized to minimize a cost function. On the other hand, blind quantum computing (BQC) has been studied to provide quantum algorithms with security by using cloud networks. A client with a limited ability to perform quantum operations hopes to have access to a quantum computer of a server, and BQC allows the client to use the server's computer without leakage of the client's information (such as input, running quantum algorithms, and output) to the server. However, BQC is designed for fault-tolerant quantum computing, and this requires many ancillary qubits, which may not be suitable for NISQ devices. Here, we propose an efficient way to implement NISQ computing with guaranteed security for the client. In our method, only  N + 1  qubits are required, under the assumption that the form of ansätze is known to the server, where  N  denotes the necessary number of the qubits in the original NISQ algorithms. The client only performs single-qubit measurements on an ancillary qubit sent from the server, and the measurement angles can specify the parameters for the ansätze of the NISQ algorithms. The no-signaling principle guarantees that neither the parameters chosen by the client nor the outputs of the algorithm are leaked to the server.    

On the approximability of random-hypergraph MAX-3-XORSAT problems with quantum algorithms

Using the prototypically hard MAX-3-XORSAT problem class, we explore the practical mechanisms for exact and approximate hardness, for both classical and quantum algorithms. We conclude that while both tasks are classically difficult for essentially the same reason, in the quantum case the mechanisms are distinct. We qualitatively identify why traditional methods (e.g. high depth QAOA) are not reliably good approximation algorithms. We propose a new method, called spectral folding, that does not suffer from these issues, and study it analytically and numerically. We show that, for random hypergraphs including extremal planted solution instances (where the ground state satisfies a much higher fraction of constraints than in truly random problems), if we define the energy to be $E = N_{unsat}-N_{sat}$, then spectral folding will return states with energy $E leq A E_{GS}$ in polynomial time, where conservatively, $A simeq 0.6$. We benchmark variations of spectral folding for random approximation-hard (planted partial solution) instances in simulation; our results support this prediction. We do not claim that this guarantee holds for all possible hypergraphs. These results suggest that quantum computers are more powerful for approximate optimization than had been previously assumed.   

Recursive Quantum Eigenvalue/Singular-value Transformation

Quantum eigenvalue transformation (QET) and quantum singular value transformation (QSVT), are versatile quantum algorithms that allow us to apply broad matrix functions to quantum states, covering quantum algorithms such as Hamiltonian simulation. However, finding a parameter set for preferable matrix functions is generally difficult: there is no analytical result other than trivial cases and we often suffer also from numerical instability. We propose recursive QET or QSVT (r-QET or r-QSVT), in which we can execute complicated matrix functions by recursively organizing block-encoding by low-degree QET or QSVT. Owing to the simplicity of recursive relations, it works only with a few parameters with exactly determining the parameters, while its iteration results in complicated matrix functions. In particular, Newton iteration allows us to analytically construct a parameter set of the matrix sign function, which can be applied for eigenstate filtering for example. The analytically-obtained parameter set composed of only different values is sufficient for executing QET of the matrix sign function with an arbitrarily small error . Our protocol will serve as an alternative protocol for constructing QET or QSVT for some useful matrix functions without numerical instability.   

Efficient Convex Unitary Decompositions for Quantum Simulation

Current quantum approaches to simulating quantum systems are still practically challenging on NISQ-era devices, because they often require extensive gate sequences and/or many ancilla qubits. We propose a hybrid quantum-classical approach to problems in quantum simulation, which we demonstrate for both standard Hamiltonian simulation and open quantum systems. Our approach generates a novel decomposition of unitary operators, which can be efficiently implemented on a quantum computer. The resulting scheme allows for resource tradeoffs between circuit depth and sample complexity, enabling near-term applications.   

Digital quantum simulations of the SSH model on a parameterized quantum circuit

Quantum computing holds immense potential to revolutionize many fields, including cryptography, material science, drug discovery, and artificial intelligence. Currently, we are in the NISQ era, where quantum devices up to a few hundred noisy qubits are available. We carry out digital quantum simulations of the non-interacting SSH model on a parameterized quantum circuit. The circuit is composed of two parts: the first part is used to prepare the initial bonding state from the product state |00....0>. The second part consists of M-layer composite quantum gates that process the quantum state. The rotation angles of the single-qubit gates are treated as variational parameters which are optimized using classical algorithms. We then study the evolutions of the ground-state energy, entanglement entropy, and mutual information on the circuit for two scenarios where the initial and final states either belong to the same or different topological phases. In the first case, we find that the ground-state energy converges exponentially fast with increasing circuit depth M. The entanglement entropy approaches saturation values after very few layers and the mutual information is nonzero only for a few close neighbor sites. However, in the second case where the initial and final state have different topologies, to prepare the ground state exactly, the minimal circuit depth is a quarter of the system size, consistent with the Lieb-Ronbinson bound for propagating the information. Additionally, the maxima of the entanglement entropy grows with the system size and mutual information propagates throughout the entire system.