Circuit Depth Research Articles

Parallelization techniques for quantum simulation of fermionic systems

Mapping fermionic operators to qubit operators is an essential step for simulating fermionic systems on a quantum computer. We investigate how the choice of such a mapping interacts with the underlying qubit connectivity of the quantum processor to enable (or impede) parallelization of the resulting Hamiltonian-simulation algorithm. It is shown that this problem can be mapped to a path coloring problem on a graph constructed from the particular choice of encoding fermions onto qubits and the fermionic interactions onto paths. The basic version of this problem is called the weak coloring problem. Taking into account the fine-grained details of the mapping yields what is called the strong coloring problem, which leads to improved parallelization performance. A variety of illustrative analytical and numerical examples are presented to demonstrate the amount of improvement for both weak and strong coloring-based parallelizations. Our results are particularly important for implementation on near-term quantum processors where minimizing circuit depth is necessary for algorithmic feasibility.

Boundaries of quantum supremacy via random circuit sampling

Google’s quantum supremacy experiment heralded a transition point where quantum computers can evaluate a computational task, random circuit sampling, faster than classical supercomputers. We examine the constraints on the region of quantum advantage for quantum circuits with a larger number of qubits and gates than experimentally implemented. At near-term gate fidelities, we demonstrate that quantum supremacy is limited to circuits with a qubit count and circuit depth of a few hundred. Larger circuits encounter two distinct boundaries: a return of a classical advantage and practically infeasible quantum runtimes. Decreasing error rates cause the region of a quantum advantage to grow rapidly. At error rates required for early implementations of the surface code, the largest circuit size within the quantum supremacy regime coincides approximately with the smallest circuit size needed to implement error correction. Thus, the boundaries of quantum supremacy may fortuitously coincide with the advent of scalable, error-corrected quantum computing.

Density-Matrix Renormalization Group Algorithm for Simulating Quantum Circuits with a Finite Fidelity

We develop a density-matrix renormalization group (DMRG) algorithm for the simulation of quantum circuits. This algorithm can be seen as the extension of time-dependent DMRG from the usual situation of hermitian Hamiltonian matrices to quantum circuits defined by unitary matrices. For small circuit depths, the technique is exact and equivalent to other matrix product state (MPS) based techniques. For larger depths, it becomes approximate in exchange for an exponential speed up in computational time. Like an actual quantum computer, the quality of the DMRG results is characterized by a finite fidelity. However, unlike a quantum computer, the fidelity depends strongly on the quantum circuit considered. For the most difficult possible circuit for this technique, the so-called "quantum supremacy" benchmark of Google Inc. , we find that the DMRG algorithm can generate bit strings of the same quality as the seminal Google experiment on a single computing core. For a more structured circuit used for combinatorial optimization (Quantum Approximate Optimization Algorithm or QAOA), we find a drastic improvement of the DMRG results with error rates dropping by a factor of 100 compared with random quantum circuits. Our results suggest that the current bottleneck of quantum computers is their fidelities rather than the number of qubits.

Observation of entanglement transition of pseudo-random mixed states

Random quantum states serve as a powerful tool in various scientific fields, including quantum supremacy and black hole physics. It has been theoretically predicted that entanglement transitions may happen for different partitions of multipartite random quantum states; however, the experimental observation of these transitions is still absent. Here, we experimentally demonstrate the entanglement transitions witnessed by negativity on a fully connected superconducting processor. We apply parallel entangling operations, that significantly decrease the depth of the pseudo-random circuits, to generate pseudo-random pure states of up to 15 qubits. By quantum state tomography of the reduced density matrix of six qubits, we measure the negativity spectra. Then, by changing the sizes of the environment and subsystems, we observe the entanglement transitions that are directly identified by logarithmic entanglement negativities based on the negativity spectra. In addition, we characterize the randomness of our circuits by measuring the distance between the distribution of output bit-string probabilities and the Porter-Thomas distribution. Our results show that superconducting processors with all-to-all connectivity constitute a promising platform for generating random states and understanding the entanglement structure of multipartite quantum systems.

Preparing maximally entangled states by monitoring the environment-system interaction

A common assumption in an open quantum system is that the noise induced by the environment, due to the interaction between a quantum system and its environment, is responsible for the disappearance of quantum properties. Interestingly, Barreiro et al. [Nature 470 (2011) 486] show, experimentally, using an open-system quantum ion traps simulator, an environment state can be engineered and controlled to pump an arbitrary quantum system toward a maximally entangled state and thus can be considered a resource for quantum information processing. Here, we demonstrate this idea to pump an arbitrary maximally mixed state into Greenberger–Horne–Zeilinger (GHZ) state by simulating our quantum circuit using one of the IBM Q processors. Barreiro et al. [Nature 470 (2011) 486] offer the circuits for the execution of the GHZ pumping. Nevertheless, those are collected gates that are suitable to the trapped-ions platform, so their circuits implementation on the IBM Q devices would result in large depth circuits and thus requires many gates. Consequently, we suggest a different circuit structure that follows a similar goal, but has been considered the features of the IBM Q platform. Moreover, we run the simulation of our circuit using the QASM simulator and free-web-based interface, IBM Quantum Experience, with and without error mitigation, to investigate the effect of the noise on the preparation of the initial mixed states of the qubits in addition to the population of the target state of the system.

A More General Quantum Credit Risk Analysis Framework

Credit risk analysis (CRA) quantum algorithms aim at providing a quadratic speedup over classical analogous methods. Despite this, experts in the business domain have identified significant limitations in the existing approaches. Thus, we proposed a new variant of the CRA quantum algorithm to address these limitations. In particular, we improved the risk model for each asset in a portfolio by enabling it to consider multiple systemic risk factors, resulting in a more realistic and complex model for each asset's default probability. Additionally, we increased the flexibility of the loss-given-default input by removing the constraint of using only integer values, enabling the use of real data from the financial sector to establish fair benchmarking protocols. Furthermore, all proposed enhancements were tested both through classical simulation of quantum hardware and, for this new version of our work, also using QPUs from IBM Quantum Experience in order to provide a baseline for future research. Our proposed variant of the CRA quantum algorithm addresses the significant limitations of the current approach and highlights an increased cost in terms of circuit depth and width. In addition, it provides a path to a substantially more realistic software solution. Indeed, as quantum technology progresses, the proposed improvements will enable meaningful scales and useful results for the financial sector.

Supervised learning of random quantum circuits via scalable neural networks

Predicting the output of quantum circuits is a hard computational task that plays a pivotal role in the development of universal quantum computers. Here we investigate the supervised learning of output expectation values of random quantum circuits. Deep convolutional neural networks (CNNs) are trained to predict single-qubit and two-qubit expectation values using databases of classically simulated circuits. These circuits are built using either a universal gate set or a continuous set of rotations plus an entangling gate, and they are represented via properly designed encodings of these gates. The prediction accuracy for previously unseen circuits is analyzed, also making comparisons with small-scale quantum computers available from the free IBM Quantum program. The CNNs often outperform these quantum devices, depending on the circuit depth, on the network depth, and on the training set size. Notably, our CNNs are designed to be scalable. This allows us exploiting transfer learning and performing extrapolations to circuits larger than those included in the training set. These CNNs also demonstrate remarkable resilience against noise, namely, they remain accurate even when trained on (simulated) expectation values averaged over very few measurements.

Efficient quantum analytic nuclear gradients with double factorization.

Efficient representations of the Hamiltonian, such as double factorization, drastically reduce the circuit depth or the number of repetitions in error corrected and noisy intermediate-scale quantum (NISQ) algorithms for chemistry. We report a Lagrangian-based approach for evaluating relaxed one- and two-particle reduced density matrices from double factorized Hamiltonians, unlocking efficiency improvements in computing the nuclear gradient and related derivative properties. We demonstrate the accuracy and feasibility of our Lagrangian-based approach to recover all off-diagonal density matrix elements in classically simulated examples with up to 327 quantum and 18 470 total atoms in QM/MM simulations with modest-sized quantum active spaces. We show this in the context of the variational quantum eigensolver in case studies, such as transition state optimization, ab initio molecular dynamics simulation, and energy minimization of large molecular systems.

Quantum Binary Field Multiplication with Optimized Toffoli Depth and Extension to Quantum Inversion

The Shor's algorithm can find solutions to the discrete logarithm problem on binary elliptic curves in polynomial time. A major challenge in implementing Shor's algorithm is the overhead of representing and performing arithmetic on binary elliptic curves using quantum circuits. Multiplication of binary fields is one of the critical operations in the context of elliptic curve arithmetic, and it is especially costly in the quantum setting. Our goal in this paper is to optimize quantum multiplication in the binary field. In the past, efforts to optimize quantum multiplication have centred on reducing the Toffoli gate count or qubits required. However, despite the fact that circuit depth is an important metric for indicating the performance of a quantum circuit, previous studies have lacked sufficient consideration for reducing circuit depth. Our approach to optimizing quantum multiplication differs from previous work in that we aim at reducing the Toffoli depth and full depth. To optimize quantum multiplication, we adopt the Karatsuba multiplication method which is based on the divide-and-conquer approach. In summary, we present an optimized quantum multiplication that has a Toffoli depth of one. Additionally, the full depth of the quantum circuit is also reduced thanks to our Toffoli depth optimization strategy. To demonstrate the effectiveness of our proposed method, we evaluate its performance using various metrics such as the qubit count, quantum gates, and circuit depth, as well as the qubits-depth product. These metrics provide insight into the resource requirements and complexity of the method. Our work achieves the lowest Toffoli depth, full depth, and the best trade-off performance for quantum multiplication. Further, our multiplication is more effective when not used in stand-alone cases. We show this effectiveness by using our multiplication to the Itoh-Tsujii algorithm-based inversion of F(x8+x4+x3+x+1).

Stochastic quantum Krylov protocol with double-factorized Hamiltonians

We propose a class of randomized quantum Krylov diagonalization (rQKD) algorithms capable of solving the eigenstate estimation problem with modest quantum resource requirements. Compared to previous real-time evolution quantum Krylov subspace methods, our approach expresses the time evolution operator ${e}^{\ensuremath{-}i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{\tau}}$ as a linear combination of unitaries and subsequently uses a stochastic sampling procedure to reduce circuit depth requirements. While our methodology applies to any Hamiltonian with fast-forwardable subcomponents, we focus on its application to the explicitly double-factorized electronic-structure Hamiltonian. To demonstrate the potential of the proposed rQKD algorithm on near-term quantum devices, we provide numerical benchmarks for a variety of molecular systems with circuit-based state-vector simulators including the effects of sampling noise, achieving ground-state energy errors of less than $1 \mathrm{kcal} {\mathrm{mol}}^{\ensuremath{-}1}$ with circuit depths orders of magnitude shallower than those required for low-rank deterministic Trotter-Suzuki decompositions.

Navigating the noise-depth tradeoff in adiabatic quantum circuits

Adiabatic quantum algorithms solve computational problems by slowly evolving a trivial state to the desired solution. On an ideal quantum computer, the solution quality improves monotonically with increasing circuit depth. By contrast, increasing the depth in current noisy computers introduces more noise and eventually deteriorates any computational advantage. What is the optimal circuit depth that provides the best solution? Here, we address this question by investigating an adiabatic circuit that interpolates between the paramagnetic and ferromagnetic ground states of the one-dimensional quantum Ising model. We characterize the quality of the final output by the density of defects $d$, as a function of the circuit depth $N$ and noise strength $\sigma$. We find that $d$ is well-described by the simple form $d_\mathrm{ideal}+d_\mathrm{noise}$, where the ideal case $d_\mathrm{ideal}\sim N^{-1/2}$ is controlled by the Kibble-Zurek mechanism, and the noise contribution scales as $d_\mathrm{noise}\sim N\sigma^2$. It follows that the optimal number of steps minimizing the number of defects goes as $\sim\sigma^{-4/3}$. We implement this algorithm on a noisy superconducting quantum processor and find that the dependence of the density of defects on the circuit depth follows the predicted non-monotonous behavior and agrees well with noisy simulations. Our work allows one to efficiently benchmark quantum devices and extract their effective noise strength $\sigma$.

Application of quantum approximate optimization algorithm to job shop scheduling problem

The Job Shop Scheduling Problem (JSSP) has always been considered as one of the most complex and industry essential scheduling problems. Optimizing the makespan of a given schedule generally involves using dedicated algorithms, local search strategies, or metaheuristics. These approaches, however, heavily rely on classical computational power, which is bounded by the physical limits of microcontrollers and power issues. Inspired by the promising results achieved for Quantum Annealing (QA) based approaches to solve JSSP instances, we propose a new approach that uses gate-model quantum architecture as an alternative to QA. We find that we can make use of the time-indexed JSSP instance representation to build a cost Hamiltonian, which can be embedded into Quantum Approximate Optimization Algorithm (QAOA) to find an optimal solution to a basic JSSP instance. We demonstrate the use of QAOA to solve the JSSP, and we evaluate its efficiency and accuracy for this problem from experimental results, as there is an increased urgency to demonstrate the applicability of quantum optimization algorithms. We also find that optimal variational parameters form patterns that can facilitate computation in bigger quantum circuits. Additionally, we compare the obtained noiseless simulation results of gate-model quantum circuits demonstrating the relationship between two evaluation criteria - makespan and energy. Finally, we analyze and present the overall performance of our approach with the increasing deadline and simulated depth of QAOA circuits.

Improved Low-Depth SHA3 Quantum Circuit for Fault-Tolerant Quantum Computers

To build a secure cryptography system in the post-quantum era, one must find the minimum security parameters against quantum attacks by estimating the quantum resources of a fault-tolerant quantum computer. In a fault-tolerant quantum computer, errors must reach an acceptable level for practical uses according to error detection and error correction processes. However, these processes utilize additional quantum resources. As the depth of the quantum circuit increases, the computation time per qubit increases together with the processing errors. Therefore, in terms of errors in quantum circuits, it is a fundamental requirement to reduce the depth by trading off the number of qubits. This paper proposes novel low-depth SHA3 quantum circuit implementations for fault-tolerant quantum computers to reduce errors. The proposed SHA3 quantum circuit was implemented with the aim of optimizing the quantum circuit depth through a trade-off between the number of qubits, the quantum gate, and the quantum depth in each function. Compared to other state-of-art techniques, the proposed method achieved T-depth and full-depth reductions of 30.3% and 80.05%, respectively. We believe that this work will contribute to the establishment of minimum security parameters for SHA3 in the quantum era.

Universal compilation for quantum state tomography

Universal compilation is a training process that compiles a trainable unitary into a target unitary. It has vast potential applications from depth-circuit compressing to device benchmarking and quantum error mitigation. Here we propose a universal compilation algorithm for quantum state tomography in low-depth quantum circuits. We apply the Fubini-Study distance as a trainable cost function and employ various gradient-based optimizations. We evaluate the performance of various trainable unitary topologies and the trainability of different optimizers for getting high efficiency and reveal the crucial role of the circuit depth in robust fidelity. The results are comparable with the shadow tomography method, a similar fashion in the field. Our work expresses the adequate capability of the universal compilation algorithm to maximize the efficiency in the quantum state tomography. Further, it promises applications in quantum metrology and sensing and is applicable in the near-term quantum computers for various quantum computing tasks.

Assisted quantum simulation of open quantum systems

Universal quantum algorithms (UQA) implemented on fault-tolerant quantum computers are expected to achieve an exponential speedup over classical counterparts. However, the deep quantum circuits make the UQA implausible in the current era. With only the noisy intermediate-scale quantum (NISQ) devices in hand, we introduce the quantum-assisted quantum algorithm, which reduces the circuit depth of UQA via NISQ technology. Based on this framework, we present two quantum-assisted quantum algorithms for simulating open quantum systems, which utilize two parameterized quantum circuits to achieve a short-time evolution. We propose a variational quantum state preparation method, as a subroutine to prepare the ancillary state, for loading a classical vector into a quantum state with a shallow quantum circuit and logarithmic number of qubits. We demonstrate numerically our approaches for a two-level system with an amplitude damping channel and an open version of the dissipative transverse field Ising model on two sites.

Calculable lower bounds on the efficiency of universal sets of quantum gates

Currently available quantum computers, so called Noisy Intermediate-Scale Quantum devices, are characterized by relatively low number of qubits and moderate gate fidelities. In such scenario, the implementation of quantum error correction is impossible and the performance of those devices is quite modest. In particular, the depth of circuits implementable with reasonably high fidelity is limited, and the minimization of circuit depth is required. Such depths depend on the efficiency of the universal set of gates used in computation, and can be bounded using the Solovay–Kitaev theorem. However, it is known that much better, asymptotically tight bounds of the form , can be obtained for specific . Those bounds are controlled by so called spectral gap, denoted . Yet, the computation of is not possible for general and in practice one considers spectral gap at a certain scale , denoted . This turns out to be sufficient to bound the efficiency of provided that one is interested in a physically feasible case, in which an error ϵ is bounded from below. In this paper we derive lower bounds on and, as a consequence, on the efficiency of universal sets of d-dimensional quantum gates satisfying an additional condition. The condition is naturally met for generic quantum gates, such as e.g. Haar random gates. Our bounds are explicit in the sense that all parameters can be determined by numerical calculations on existing computers, at least for small d. This is in contrast with known lower bounds on which involve parameters with ambiguous values.

Revisiting semiconductor bulk hamiltonians using quantum computers

With the advent of near-term quantum computers, it is now possible to simulate solid-state properties using quantum algorithms. By an adequate description of the system's Hamiltonian, variational methods enable to fetch of the band structure and other fundamental properties as transition probabilities. Here, we describe semiconductor structures of the III-V family using k·p Hamiltonians and obtain their band structures using a state vector solver, a probabilistic simulator, and a real noisy-device simulator. The resulting band structures are in good agreement with those obtained by direct diagonalization of the Hamiltonian. The simulation times depend on the optimizer, circuit depth, and simulator used. Finally, with the optimized eigenstates, we convey the inter-band absorption probability, demonstrating the possibility of analyzing the fundamental properties of crystalline systems using quantum computers.

Amplitude Estimation from Quantum Signal Processing

Amplitude estimation algorithms are based on Grover's algorithm: alternating reflections about the input state and the desired outcome. But what if we are given the ability to perform arbitrary rotations, instead of just reflections? In this situation, we find that quantum signal processing lets us estimate the amplitude in a more flexible way. We leverage this technique to give improved and simplified algorithms for many amplitude estimation tasks: we perform non-destructive estimation without any assumptions on the amplitude, develop an algorithm with improved performance in practice, present a new method for unbiased amplitude estimation, and finally give a simpler method for trading quantum circuit depth for more repetitions of short circuits.

Description of reaction and vibrational energetics of CO2–NH3 interaction using quantum computing algorithms

CO2 capture is critical to solving global warming. Amine-based solvents are extensively used to chemically absorb CO2. Thus, it is crucial to study the chemical absorption of CO2 by amine-based solvents to better understand and optimize CO2 capture processes. Here, we use quantum computing algorithms to quantify molecular vibrational energies and reaction pathways between CO2 and a simplified amine-based solvent model—NH3. Molecular vibrational properties are important to understanding kinetics of reactions. However, the molecule size correlates with the strength of anharmonicity effect on vibrational properties, which can be challenging to address using classical computing. Quantum computing can help enhance molecular vibrational calculations by including anharmonicity. We implement a variational quantum eigensolver (VQE) algorithm in a quantum simulator to calculate ground state vibrational energies of reactants and products of the CO2 and NH3 reaction. The VQE calculations yield ground vibrational energies of CO2 and NH3 with similar accuracy to classical computing. In the presence of hardware noise, Compact Heuristic for Chemistry (CHC) ansatz with shallower circuit depth performs better than Unitary Vibrational Coupled Cluster. The “Zero Noise Extrapolation” error-mitigation approach in combination with CHC ansatz improves the vibrational calculation accuracy. Excited vibrational states are accessed with quantum equation of motion method for CO2 and NH3. Using quantum Hartree–Fock (HF) embedding algorithm to calculate electronic energies, the corresponding reaction profile compares favorably with Coupled Cluster Singles and Doubles while being more accurate than HF. Our research showcases quantum computing applications in the study of CO2 capture reactions.

Evolution strategies: application in hybrid quantum-classical neural networks

With the rapid development of quantum computers, several applications are being proposed for them. Quantum simulations, simulation of chemical reactions, solution of optimization problems and quantum neural networks (QNNs) are some examples. However, problems such as noise, limited number of qubits and circuit depth, and gradient vanishing must be resolved before we can use them to their full potential. In the field of quantum machine learning, several models have been proposed. In general, in order to train these different models, we use the gradient of a cost function with respect to the model parameters. In order to obtain this gradient, we must compute the derivative of this function with respect to the model parameters. One of the most used methods in the literature to perform this task is the parameter-shift rule method. This method consists of evaluating the cost function twice for each parameter of the QNN. A problem with this method is that the number of evaluations grows linearly with the number of parameters. In this work, we study an alternative method, called evolution strategies (ES), which are a family of black box optimization algorithms which iteratively update the parameters using a search gradient. An advantage of the ES method is that in using it, one can control the number of times the cost function will be evaluated. We apply the ES method to the binary classification task, showing that this method is a viable alternative for training QNNs. However, we observe that its performance will be strongly dependent on the hyperparameters used. Furthermore, we also observe that this method, alike the parameter shift rule method, suffers from the problem of gradient vanishing.