In recent years, rapid advancements in quantum technologies have resulted in heightened research activity, particularly in the domain of efficient implementation of computationally intensive kernels from prominent quantum algorithms. This is partly addressed by increasingly sophisticated synthesis techniques for quantum circuits. In this paper, we study the efficient Clifford +T circuit realization of two kernels, namely square root and inverse square root for different number formats. Quantum algorithms for solving Toeplitz system of linear equations, numerical methods for solving Poisson equation and in general for scientific computing require such kernels with customizable trade-off points for accuracy and performance. We present an efficient circuit design, obtained via detailed exploration of design choices in terms of accuracy and quantum costs. The manual circuits designed provide upto $50 \times $ reduction in qubit count for high precision circuits. Furthermore, a detailed benchmarking of state-of-the-art synthesis tools is undertaken for these kernels along with detailed comparison against manually-implemented designs.
In recent years, rapid advancements in quantum technologies have resulted in heightened research activity, particularly in the domain of efficient implementation of computationally intensive kernels from prominent quantum algorithms. This is partly addressed by increasingly sophisticated synthesis techniques for quantum circuits. In this paper, we study the efficient Clifford +T circuit realization of two kernels, namely square root and inverse square root for different number formats. Quantum algorithms for solving Toeplitz system of linear equations, numerical methods for solving Poisson equation and in general for scientific computing require such kernels with customizable trade-off points for accuracy and performance. We present an efficient circuit design, obtained via detailed exploration of design choices in terms of accuracy and quantum costs. The manual circuits designed provide upto 50× reduction in qubit count for high precision circuits. Furthermore, a detailed benchmarking of state-of-the-art synthesis tools is undertaken for these kernels along with detailed comparison against manually-implemented designs.
Quantum computers offer the potential to extend our abilities to tackle computational problems in fields such as number theory, encryption, search and scientific computation. Up to a superpolynomial speedup has been reported for quantum algorithms in these areas. Motivated by the promise of faster computations, the development of quantum machines has caught the attention of both academics and industry researchers. Quantum machines are now at sizes where implementations of quantum algorithms or their components are now becoming possible. In order to implement quantum algorithms on quantum machines, resource efficient circuits and functional blocks must be designed. In this work, we propose quantum circuits for Galois and integer arithmetic. These quantum circuits are necessary building blocks to realize quantum algorithms. The design of resource efficient quantum circuits requires the designer takes into account the gate cost, quantum bit (qubit) cost, depth and garbage outputs of a quantum circuit. Existing quantum machines do not have many qubits meaning that circuits with high qubit cost cannot be implemented. In addition, quantum circuits are more prone to errors and garbage output removal adds to overall cost. As more gates are used, a quantum circuit sees an increased rate of failure. Failures and error rates can be countered by using quantum error correcting codes and fault tolerant implementations of universal gate sets (such as Clifford+T gates). However, Clifford+T gates are costly to implement with the T gate being significantly more costly than the Clifford gates. As a result, designers working with Clifford+T gates seek to minimize the number of T gates (T-count) and the depth of T gates (T-depth). In this work, we propose quantum circuits for Galois and integer arithmetic with lower T-count, T-depth and qubit cost than existing work. This work presents novel quantum circuits for squaring and exponentiation over binary extension fields (Galois fields of form GF(2 m )). The proposed circuits are shown to have lower depth, qubit and gate cost to existing work. We also present quantum circuits for the core operations of multiplication and division which enjoy lower T-count, T-depth and qubit costs compared to existing work. This work also illustrates the design of a T-count and qubit cost efficient design for the square root. This work concludes with an illustration of how the arithmetic circuits can be combined into a functional block to implement quantum image processing algorithms.
Quantum square root operation is a useful building block in implementing important quantum algorithms such as the Harrow–Hassidim–Lloyd (HHL) algorithm, algorithms in physics and quantum chemistry, and also used in computing roots of quadratic equation, evaluating quadratic congruence, solving Hamiltonian and Poisson equations, trigonometric functions, computing logarithm and fractional exponent, and in quantum chemistry [1]–[5]. Quantum circuits could be made fault-tolerant by using error correcting codes and fault-tolerant quantum gates (such as the Clifford + T-gates). However, The T-gates are expensive to implement than other fault-tolerant gates (such as Clifford+T gates) [6] [7]. Therefore, quantum circuits with low T-count and qubit cost is preferred. In this paper, we present a novel quantum square root architecture optimized for T-count and Qubit cost that produces no garbage outputs. The proposed square root circuit accepts integer and fractional inputs. We use the Babylonian method to calculate the square root. To reduce the number of iteration, the initial guess required in the Babylonian method is calculated from the integer square root circuit in [8] which reduces the cost measures. To further save the cost, we use the resource efficient non-restoring division circuit shown in [9] and quantum adder shown in [6]. The proposed quantum square root circuit sees an asymptotic reduction of 95.84% in T-count and 88.89% in Qubit cost with respect to Bhaskar et al. [1]. With respect to Dutta et al. [10] the design sees an asymptotic reduction of 96% in T-count, and 88.89% in Qubit cost.
Quantum algorithms for scientific computing require modules implementing fundamental functions, such as the square root, the logarithm, and others. We require algorithms that have a well-controlled numerical error, that are uniformly scalable and reversible (unitary), and that can be implemented efficiently. We present quantum algorithms and circuits for computing the square root, the natural logarithm, and arbitrary fractional powers. We provide performance guarantees in terms of their worst-case accuracy and cost. We further illustrate their performance by providing tests comparing them to the respective floating point implementations found in widely used numerical software.
Quantum algorithms for scientific computing require modules implementing fundamental functions, such as the square root, the logarithm, and others. We require algorithms that have a well-controlled numerical error, that are uniformly scalable and reversible (unitary), and that can be implemented efficiently. We present quantum algorithms and circuits for computing the square root, the natural logarithm, and arbitrary fractional powers. We provide performance guarantees in terms of their worst-case accuracy and cost. We further illustrate their performance by providing tests comparing them to the respective floating point implementations found in widely used numerical software.
Resource-efficient and high-precision approximate synthesis of quantum circuits expressed in the Clifford+T gate set is vital for Fault-Tolerant quantum computing.Efficient optimal methods are known for single-qubit R Z unitaries, otherwise the problem is generally intractable.Search-based methods, like simulated annealing, empirically generate low resource cost approximate implementations of general multi-qubit unitaries so long as low precision (Hilbert-Schmidt distances of ε ≥ 10 -2 ) can be tolerated.These algorithms build up circuits that directly invert target unitaries.We instead leverage search-based methods to first approximately diagonalize a unitary, then perform the inversion analytically.This lets difficult continuous rotations be bypassed and handled in a postprocessing step.Our approach improves both the implementation precision and run time of synthesis algorithms by orders of magnitude when evaluated on unitaries from real quantum algorithms.On benchmarks previously synthesizable only with analytical techniques like the Quantum Shannon Decomposition, diagonalization uses an average of 95% fewer non-Clifford gates.
Quantum circuits for basic mathematical functions such as the square root are required to implement scientific computing algorithms on quantum computers. Quantum circuits that are based on Clifford+T gates can easily be made fault tolerant, but the T gate is very costly to implement. As a result, reducing T-count has become an important optimization goal. Further, quantum circuits with many qubits are difficult to realize, making designs that save qubits and produce no garbage outputs desirable. In this work, we present a T-count optimized quantum square root circuit with only 2 ṡ n + 1 qubits and no garbage output. To make a fair comparison against existing work, the Bennett’s garbage removal scheme is used to remove garbage output from existing works. We determined that our proposed design achieves an average T-count savings of 43.44%, 98.95%, 41.06%, and 20.28% as well as qubit savings of 85.46%, 95.16%, 90.59%, and 86.77% compared to existing works.