Quantum computing promises to provide exponential speed-ups to certain classes of problems. In many such quantum algorithms, a classical vector $\mathbf{b}$ is encoded in the amplitudes of a quantum state $| b\rangle$. However, efficient amplitude encoding of an arbitrary vector $\mathbf{b}$ in $| b \rangle$ is known to be a difficult problem that can reverse the gains of such algorithms. More importantly, an arbitrary quantum state of $Q$ qubits generally requires $\sim 2^Q$ number of entangling gates, which is problematic for today's Noisy-Intermediate Scale Quantum (NISQ) computers where large numbers of such error-prone gates will result in significant decoherence before initialization is completed. In this work, we demonstrate a deterministic and scalable initialization algorithm that allows for states with low entanglement to be initialized on actual quantum computers with more than an order of magnitude less entangling gates and significantly shallower circuits as compared to isometric decomposition. For states with higher entanglement, optimally approximate states with lower entanglement may be found and initialized with similar performance. We show this to be true for various cases of interest such as the normal and log-normal distributions. Unlike variational approaches, our method requires no classical optimizations in high dimensional spaces and is not plagued with exponentially vanishing gradients or a multitude of local minimas.
Quantum computing promises to provide exponential speed-ups to certain classes of problems. In many such quantum algorithms, a classical vector b is encoded in the amplitudes of a quantum state ∣b⟩. However, efficient amplitude encoding of an arbitrary vector b in ∣b⟩ is known to be a difficult problem that can reverse the gains of such algorithms. More importantly, an arbitrary quantum state of Q qubits generally requires ∼2Q number of entangling gates, which is problematic for today's Noisy-Intermediate Scale Quantum (NISQ) computers where large numbers of such error-prone gates will result in significant decoherence before initialization is completed. In this work, we demonstrate a deterministic and scalable initialization algorithm that allows for states with low entanglement to be initialized on actual quantum computers with more than an order of magnitude less entangling gates and significantly shallower circuits as compared to isometric decomposition. For states with higher entanglement, optimally approximate states with lower entanglement may be found and initialized with similar performance. We show this to be true for various cases of interest such as the normal and log-normal distributions. Unlike variational approaches, our method requires no classical optimizations in high dimensional spaces and is not plagued with exponentially vanishing gradients or a multitude of local minimas.
Quantum computing promises to provide exponential speed-ups to certain classes of problems. In many such algorithms, a classical vector b is encoded in the amplitudes of a quantum state ∣b⟩. However, efficiently preparing ∣b⟩ is known to be a difficult problem because an arbitrary state of Q qubits generally requires approximately 2Q entangling gates, which results in significant decoherence on today's noisy intermediate-scale quantum (NISQ) computers. We present a deterministic (nonvariational) algorithm that allows us to flexibly reduce the quantum resources required for state preparation in an entanglement-efficient manner. Although this comes at the expense of reduced theoretical fidelity, actual fidelities on current NISQ computers might actually be higher due to reduced decoherence. We show this to be true for various cases of interest, such as the normal and log-normal distributions. For low-entanglement states, our algorithm can prepare states with more than an order of magnitude fewer entangling gates as compared with isometric decomposition.
Quantum State Preparation Algorithms consist of defining a sequence or unitary operations to load a specific target state on a quantum computer. We can use those algorithms in appli- cations such as quantum machine learning. However, some state preparation algorithms have exponential circuit complexity with the number of qubits on the system. That is the case of amplitude encoding algorithms, which is an encoding type for loading normalized data into the probability amplitudes of the state. To circumvent this overhead in circuits’ complexity, works explore specific properties of quantum states to optimize the circuit’s complexity, such as sparsity or symmetry. Other works explore simplifying the quantum circuit to load an ap- proximate quantum state. It is the case of Quantum Generative Adversarial Networks, which use a specific circuit architecture comprised of alternating blocks of single-qubit rotations and two-qubit entangling controlled gates. But when trained to load random distributions on, we observed the performance deteriorates as the number of qubits increases in terms of relative entropy. In this work, we propose different architectures for the Quantum Generative mod- els based on the state preparation algorithm known as Low-Rank. Through experiments for loading the log-normal distribution, we show error reductions in quantum state initialization.
Abstract Quantum algorithms and simulations often require the preparation of complex states through sequences of 2-qubit gates. For a generic quantum state, the number of required gates grows exponentially with the number of qubits, becoming unfeasible on near-term quantum devices. Here, we aim at creating an approximate encoding of the target state using a limited number of gates. As a first step, we consider a quantum state that is efficiently represented classically, such as a one-dimensional matrix product state. Using tensor network techniques, we develop and implement an efficient optimization algorithm that approaches the optimal implementation, requiring a polynomial number of iterations. We, next, consider the implementation of the proposed optimization algorithm directly on a quantum computer and overcome inherent barren plateaus by employing a local cost function. Our work offers a universal method to prepare target states using local gates and represents a significant improvement over known strategies.
Quantum computing not only holds the potential to solve long-standing problems in quantum physics, but also to offer speed-ups across a broad spectrum of other fields. Access to a computational space that incorporates quantum effects, such as superposition and entanglement, enables the derivation of promising quantum algorithms for important tasks, including preparing the ground state of a quantum system or predicting its evolution over time. Successfully tackling these tasks promises insights into significant theoretical and technological questions, such as superconductivity and the design of new materials. The aim of quantum algorithms is to use a series of quantum operations in a quantum circuit to solve a problem beyond the reach of classical computers. However, the noise and limited scale of current quantum computers restricts these circuits to moderate sizes and depths. As a result, many prominent algorithms are currently infeasible to run for problem sizes of practical interest. In response, recent research focused on variational quantum algorithms, which allow the selection of circuits that act within a quantum device's capabilities. Yet, these algorithms can require the execution of a large number of circuits, leading to prohibitively long computation times. This doctoral thesis develops two main techniques to reduce these quantum computational resource requirements, with the goal of scaling up application sizes on current quantum processors. The first approach is based on stochastic approximations of computationally costly quantities, such as quantum circuit gradients or the quantum geometric tensor (QGT). The second method takes a different perspective on the QGT, leading to a potentially more efficient description of time evolution on current quantum computers. Both techniques rely on maintaining available information and only computing necessary corrections, instead of re-computing possibly redundant data. The main focus of application for our algorithms is the simulation of quantum systems, broadly defined as including the preparation of ground and thermal states, and the real- and imaginary-time propagation of a system. The developed subroutines, however, can further be utilized in the fields of optimization or machine learning. Our algorithms are benchmarked on a range of representative models, such as Ising or Heisenberg spin models, both in numerical simulations and experiments on the hardware. In combination with error mitigation techniques, the latter is scaled up to 27 qubits; into a regime that variational quantum algorithms are challenging to scale to on noisy quantum computers without our algorithms.
Gate-based quantum algorithms are typically implemented by circuits consisting of many single-qubit and multi-qubit gates operating on a quantum input state, followed by measurements of the output state. For certain quantum subroutines, such as initial state preparation and quantum Fourier transforms, explicit decompositions of the circuit in terms of single-qubit and two-qubit maximally entangling gates may exist. However, they often lead to large-depth circuits that are challenging for noisy intermediate-scale quantum (NISQ) hardware. Additionally, exact decompositions might only exist for some modular quantum circuits. Therefore, it is essential to find gate combinations that approximate these circuits to high fidelity with potentially low depth. Gradient-based optimization has been used to find such approximate decompositions. Still, these traditional optimizers often run into problems of slow convergence requiring many iterations, and performing poorly in the presence of noise, a factor that is especially relevant for NISQ hardware. Here we present iteratively preconditioned gradient descent (IPG) for optimizing quantum circuits and demonstrate performance speedups for state preparation and implementation of quantum algorithmic subroutines. IPG is a noise-resilient, higher-order algorithm that has shown promising gains in convergence speed for classical optimizations, converging locally at a linear rate for convex problems and superlinearly when the solution is unique. Specifically, we show an improvement in fidelity by a factor of 104for preparing a 4-qubit W state and a maximally entangled 5-qubit GHZ state with compared to other commonly used classical optimizers tuning the same ansatz. Such faster convergence with promise for noise-resilience could provide advantages for quantum algorithms on NISQ hardware, especially since the cost of running each iteration on a quantum computer is substantially higher than the classical optimizer step.
In the noisy intermediate-scale quantum (NISQ) era, distributed quantum computation has garnered considerable interest, as it overcomes the physical limitations of single-device architectures and enables scalable quantum information processing. In this study, we focus on the challenge of achieving exact amplitude amplification for quantum states with arbitrary amplitude distributions and subsequently propose a Distributed Exact Quantum Amplitude Amplification Algorithm (DEQAAA). Specifically, (1) it supports partitioning across any number of nodes t within the range 2≤t≤n; (2) the maximum qubit count required for any single node is expressed as $\max \left(n_0,n_1,\dots,n_{t-1} \right) $, where nj represents the number of qubits at the j-th node, with ∑j=0t−1nj=n; (3) it can realize exact amplitude amplification for multiple targets of a quantum state with arbitrary amplitude distributions; (4) we verify the effectiveness of DEQAAA by resolving a specific exact amplitude amplification task involving two targets (8 and 14 in decimal) via MindSpore Quantum, a quantum simulation software, with tests conducted on 4-qubit, 6-qubit, 8-qubit and 10-qubit systems. Notably, through the decomposition of Cn−1PS gates, DEQAAA demonstrates remarkable advantages in both quantum gate count and circuit depth as the qubit number scales, thereby boosting its noise resilience. In the 10-qubit scenario, for instance, it achieves a reduction of over 97% in both indicators compared to QAAA and EQAAA, underscoring its outstanding resource-saving performance.