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arXiv:2406.04004 · IEEE International Conference on Quantum Software (QSW)

T-Count Optimizing Genetic Algorithm for Quantum State Preparation

Andrew L. Wright, Marco Lewis, Paolo Zuliani, Sadegh Soudjani

Published Jul 7, 2024• 1 citation
arXivHTMLPDF

Abstract

Quantum state preparation is a crucial process within numerous quantum algorithms, and the need for efficient initialization of quantum registers is ever increasing as demand for useful quantum computing grows. The problem arises as the number of qubits to be initialized grows, the circuits required to implement the desired state also exponentially increase in size leading to loss of fidelity to noise. This is mainly due to the susceptibility to environmental effects of the non-Clifford T gate, whose use should thus be reduced as much as possible. In this paper, we present and utilize a genetic algorithm for state preparation circuits consisting of gates from the Clifford + T gate set and optimize them in T-Count as to reduce the impact of noise. Whilst the method presented here does not always produce the most accurate circuits in terms of fidelity, it can generate high-fidelity, non-trivial quantum states such as quantum Fourier transform states. In addition, our algorithm does automatically generate fault tolerantly implementable solutions where the number of the most error prone components is reduced. We present an evaluation of the algorithm when trialed against preparing random, Poisson probability distribution, W, GHZ, and quantum Fourier transform states. We also experimentally demonstrate the scalability issues as qubit count increases, which highlights the need for further optimization of the search process.

DOI
10.1109/qsw62656.2024.00020

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Abstract An important part of reaping computational advantage from a quantum computer is to reduce the quantum resources needed to implement a desired quantum algorithm. Quantum algorithms that are too large to be practical on noisy intermediate scale quantum devices will require fault-tolerant error correction. This work focuses on reducing the physical cost of implementing quantum algorithms when using the state-of-the-art fault-tolerant quantum error correcting codes, in particular, those for which implementing the T gate consumes vastly more resources than the other gates in the gate set. More specifically, in this paper we consider the group of unitaries that can be exactly implemented by a quantum circuit consisting of the Clifford + T gate set. The Clifford + T gate set is a universal gate set and in this group, using state-of-the-art surface codes, the T gate is by far the most expensive component to implement fault-tolerantly. So it is important to minimize the number of T gates necessary for a fault-tolerant implementation. Our primary interest is to compute a circuit for a given n -qubit unitary U , using the minimum possible number of T gates (called the T-count of unitary U ). We consider the problem COUNT-T, the optimization version of which aims to find the T-count of U . In its decision version the goal is to decide if the T-count is at most some positive integer m . Given an oracle for COUNT-T, we can compute a T-count-optimal circuit in time polynomial in the T-count and dimension of U . We give a provable classical algorithm that solves COUNT-T (decision) in time O N 2 ( c − 1 ) ⌈ m c ⌉ p o l y ( m , N ) and space O N 2 ⌈ m c ⌉ p o l y ( m , N ) , where N = 2 n and c ⩾ 2. This gives a space-time trade-off for solving this problem with variants of meet-in-the-middle techniques. We also introduce an asymptotically faster multiplication method that shaves a factor of N 0.7457 off of the overall complexity. Lastly, beyond our improvements to the rigorous algorithm, we give a heuristic algorithm that outputs a T-count-optimal circuit and has space and time complexity poly( m , N ), under some assumptions. In our heuristic algorithm we developed a novel way of pruning the search space. While our heuristic method still scales exponentially with the number of qubits (though with a lower exponent), there is a large improvement by going from exponential to polynomial scaling with m . We implemented our heuristic algorithm with up to 4 qubit unitaries and obtained a significant improvement in time. For all benchmark and random unitaries we studied, the T-count returned by our algorithm is at most the T-count of their circuits shown in previous papers.

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Any quantum computation begins with some process of state preparation where the prepared state functions as the input value for the computation. For most states, preparation becomes exceedingly more demanding as the number of qubits increases. Moreover, the noise present in the gate operations further complicates the process by contributing to the decoherence of the state. This thesis studies possible approaches to address these problems. For a reader not familiar with the basic principles of quantum computing, Chapter 2 should provide a sufficient summary in order to follow through the rest of the thesis. In Chapter 3 a standard method of state preparation is presented in detail for 2 and 4-qubit cases and then generalized for an arbitrary number of qubits. However in order to present a comprehensive coverage, the necessary gate decompositions for state preparation are first presented and proved. The number of required quantum gates for the preparation of a general state increases exponentially in the number of qubits. Each gate introduces a small error in the computation, the cumulative effect of which can make the decoherence in the final state very severe. Therefore it is perhaps more suitable to prepare a less complex approximation of the target state with fewer gates to obtain better fidelity in the presence of noise. For this purpose, two methods are presented: low-rank state preparation algorithm (LRSP) and a genetic algorithm. LRSP provides a systematic way to generate circuits for approximate state preparation; however, for any given state LRSP can make only very specific approximations, giving us a very limited range of circuits to work with. For this reason our work introduces a genetic algorithm to iterate over the LRSP circuits and to provide a wider range of candidate circuits for approximate state preparation. Using this approach we prepared a range of approximate state preparation circuits for 1000 different Haar-random states and ran them with simulated noise on IBM Qiskit FakeVigo backend. We found that the numerical results matched closely with our theoretical prediction and showed that when run on a noisy quantum computer, the highest fidelity is obtained not for the exact state preparation circuit but for a shorter approximate circuit.

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Exploring the optimality of approximate state preparation quantum circuits with a genetic algorithm

Tom Rindell, Berat Yenilen, Niklas Halonen +3 more

We study the approximate state preparation problem on noisy intermediate-scale quantum (NISQ) computers by applying a genetic algorithm to generate quantum circuits for state preparation. The algorithm can account for the specific characteristics of the physical machine in the evaluation of circuits, such as the native gate set and qubit connectivity. We use our genetic algorithm to optimize the circuits provided by the low-rank state preparation algorithm introduced by Araujo et al., and find substantial improvements to the fidelity in preparing Haar random states with a limited number of CNOT gates. Moreover, we observe that already for a 5-qubit quantum processor with limited qubit connectivity and significant noise levels (IBM Falcon 5T), the maximal fidelity for Haar random states is achieved by a short approximate state preparation circuit instead of the exact preparation circuit. We also present a theoretical analysis of approximate state preparation circuit complexity to motivate our findings. Our genetic algorithm for quantum circuit discovery is freely available at https://github.com/beratyenilen/qc-ga .

arXiv:2210.06411•Oct 12, 2022•1 citation
Quantum state preparation is a crucial process within numerous quantum algorithms, and the need for efficient initialization of quantum registers is ever increasing as demand for useful quantum computing grows. The problem arises as the number of qubits to be initialized grows, the circuits required to implement the desired state also exponentially increase in size leading to loss of fidelity to noise. This is mainly due to the susceptibility to environmental effects of the non-Clifford T gate, whose use should thus be reduced as much as possible. In this paper, we present and utilize a genetic algorithm for state preparation circuits consisting of gates from the Clifford + T gate set and optimize them in T-Count as to reduce the impact of noise. Whilst the method presented here does not always produce the most accurate circuits in terms of fidelity, it can generate high-fidelity, non-trivial quantum states such as quantum Fourier transform states. In addition, our algorithm does automatically generate fault tolerantly implementable solutions where the number of the most error prone components is reduced. We present an evaluation of the algorithm when trialed against preparing random, Poisson probability distribution, W, GHZ, and quantum Fourier transform states. We also experimentally demonstrate the scalability issues as qubit count increases, which highlights the need for further optimization of the search process.
The efficient preparation of quantum states is an important step in the execution of many quantum algorithms. In the noisy intermediate-scale quantum (NISQ) computing era, this is a significant challenge given quantum resources are scarce and typically only low-depth quantum circuits can be implemented on physical devices. We present a genetic algorithm for state preparation (GASP) which generates relatively low-depth quantum circuits for initialising a quantum computer in a specified quantum state. The method uses a basis set of [Formula: see text], [Formula: see text], [Formula: see text], and CNOT gates and a genetic algorithm to systematically generate circuits to synthesize the target state to the required fidelity. GASP can produce more efficient circuits of a given accuracy with lower depth and gate counts than other methods. This variability of the required accuracy facilitates overall higher accuracy on implementation, as error accumulation in high-depth circuits can be avoided. We directly compare the method to the state initialisation technique based on an exact synthesis technique by implemented in IBM Qiskit simulated with noise and implemented on physical IBM Quantum devices. Results achieved by GASP outperform Qiskit's exact general circuit synthesis method on a variety of states such as Gaussian states and W-states, and consistently show the method reduces the number of gates required for the quantum circuits to generate these quantum states to the required accuracy.
The efficient preparation of quantum states is an important step in the execution of many quantum algorithms. In the noisy intermediate-scale quantum (NISQ) computing era, this is a significant challenge given quantum resources are scarce and typically only low-depth quantum circuits can be implemented on physical devices. We present a genetic algorithm for state preparation (GASP) which generates relatively low-depth quantum circuits for initialising a quantum computer in a specified quantum state. The method uses a basis set of R_x, R_y, R_z, and CNOT gates and a genetic algorithm to systematically generate circuits to synthesize the target state to the required fidelity. GASP can produce more efficient circuits of a given accuracy with lower depth and gate counts than other methods. This variability of the required accuracy facilitates overall higher accuracy on implementation, as error accumulation in high-depth circuits can be avoided. We directly compare the method to the state initialisation technique based on an exact synthesis technique by implemented in IBM Qiskit simulated with noise and implemented on physical IBM Quantum devices. Results achieved by GASP outperform Qiskit's exact general circuit synthesis method on a variety of states such as Gaussian states and W-states, and consistently show the method reduces the number of gates required for the quantum circuits to generate these quantum states to the required accuracy.
Quantum Error Correction (QEC) is the corner-stone of practical Fault-Tolerant Quantum Computing (FTQC), but incurs enormous resource overheads. Circuits must decompose into Clifford +T gates, and the non-transversal T gates demand costly magic-state distillation. As circuit complexity grows, sequential T-gate layers (” T-depth”) increase, amplifying the spatiotemporal overhead of QEC. Optimizing T-depth is NP-hard, and existing greedy or brute-force strategies are either inefficient or computationally prohibitive. We frame T-depth reduction as a search optimization problem and present a Genetic Algorithm (GA) framework that approximates optimal layer-merge patterns across the non-convex search space. We introduce a mathematical formulation of the circuit expansion for systematic layer reordering and a greedy initial merge-pair selection, accelerating the convergence and enhancing the solution quality. In our benchmark with ∼90−100\mathbf{\sim 9 0 - 1 0 0}∼90−100 qubits, our method reduces T-depth by 79.23% and overall T-count by 41.86%. Compared to the standard reversible circuit benchmarks, we achieve a ∼2.58×\sim 2.58 \times∼2.58× average improvement in T-depth over the state-of-the-art methods, demonstrating its viability for near-term FTQC.
Abstract An important part of reaping computational advantage from a quantum computer is to reduce the quantum resources needed to implement a desired quantum algorithm. Quantum algorithms that are too large to be practical on noisy intermediate scale quantum devices will require fault-tolerant error correction. This work focuses on reducing the physical cost of implementing quantum algorithms when using the state-of-the-art fault-tolerant quantum error correcting codes, in particular, those for which implementing the T gate consumes vastly more resources than the other gates in the gate set. More specifically, in this paper we consider the group of unitaries that can be exactly implemented by a quantum circuit consisting of the Clifford + T gate set. The Clifford + T gate set is a universal gate set and in this group, using state-of-the-art surface codes, the T gate is by far the most expensive component to implement fault-tolerantly. So it is important to minimize the number of T gates necessary for a fault-tolerant implementation. Our primary interest is to compute a circuit for a given n -qubit unitary U , using the minimum possible number of T gates (called the T-count of unitary U ). We consider the problem COUNT-T, the optimization version of which aims to find the T-count of U . In its decision version the goal is to decide if the T-count is at most some positive integer m . Given an oracle for COUNT-T, we can compute a T-count-optimal circuit in time polynomial in the T-count and dimension of U . We give a provable classical algorithm that solves COUNT-T (decision) in time O N 2 ( c − 1 ) ⌈ m c ⌉ p o l y ( m , N ) and space O N 2 ⌈ m c ⌉ p o l y ( m , N ) , where N = 2 n and c ⩾ 2. This gives a space-time trade-off for solving this problem with variants of meet-in-the-middle techniques. We also introduce an asymptotically faster multiplication method that shaves a factor of N 0.7457 off of the overall complexity. Lastly, beyond our improvements to the rigorous algorithm, we give a heuristic algorithm that outputs a T-count-optimal circuit and has space and time complexity poly( m , N ), under some assumptions. In our heuristic algorithm we developed a novel way of pruning the search space. While our heuristic method still scales exponentially with the number of qubits (though with a lower exponent), there is a large improvement by going from exponential to polynomial scaling with m . We implemented our heuristic algorithm with up to 4 qubit unitaries and obtained a significant improvement in time. For all benchmark and random unitaries we studied, the T-count returned by our algorithm is at most the T-count of their circuits shown in previous papers.
Any quantum computation begins with some process of state preparation where the prepared state functions as the input value for the computation. For most states, preparation becomes exceedingly more demanding as the number of qubits increases. Moreover, the noise present in the gate operations further complicates the process by contributing to the decoherence of the state. This thesis studies possible approaches to address these problems. For a reader not familiar with the basic principles of quantum computing, Chapter 2 should provide a sufficient summary in order to follow through the rest of the thesis. In Chapter 3 a standard method of state preparation is presented in detail for 2 and 4-qubit cases and then generalized for an arbitrary number of qubits. However in order to present a comprehensive coverage, the necessary gate decompositions for state preparation are first presented and proved. The number of required quantum gates for the preparation of a general state increases exponentially in the number of qubits. Each gate introduces a small error in the computation, the cumulative effect of which can make the decoherence in the final state very severe. Therefore it is perhaps more suitable to prepare a less complex approximation of the target state with fewer gates to obtain better fidelity in the presence of noise. For this purpose, two methods are presented: low-rank state preparation algorithm (LRSP) and a genetic algorithm. LRSP provides a systematic way to generate circuits for approximate state preparation; however, for any given state LRSP can make only very specific approximations, giving us a very limited range of circuits to work with. For this reason our work introduces a genetic algorithm to iterate over the LRSP circuits and to provide a wider range of candidate circuits for approximate state preparation. Using this approach we prepared a range of approximate state preparation circuits for 1000 different Haar-random states and ran them with simulated noise on IBM Qiskit FakeVigo backend. We found that the numerical results matched closely with our theoretical prediction and showed that when run on a noisy quantum computer, the highest fidelity is obtained not for the exact state preparation circuit but for a shorter approximate circuit.
We study the approximate state preparation problem on noisy intermediate-scale quantum (NISQ) computers by applying a genetic algorithm to generate quantum circuits for state preparation. The algorithm can account for the specific characteristics of the physical machine in the evaluation of circuits, such as the native gate set and qubit connectivity. We use our genetic algorithm to optimize the circuits provided by the low-rank state preparation algorithm introduced by Araujo et al., and find substantial improvements to the fidelity in preparing Haar random states with a limited number of CNOT gates. Moreover, we observe that already for a 5-qubit quantum processor with limited qubit connectivity and significant noise levels (IBM Falcon 5T), the maximal fidelity for Haar random states is achieved by a short approximate state preparation circuit instead of the exact preparation circuit. We also present a theoretical analysis of approximate state preparation circuit complexity to motivate our findings. Our genetic algorithm for quantum circuit discovery is freely available at https://github.com/beratyenilen/qc-ga .