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arXiv:2401.05973

Geodesic Algorithm for Unitary Gate Design with Time-Independent Hamiltonians

Lewis, Dylan, Roeland Wiersema, Juan Carrasquilla, Sougato Bose

Published Jan 11, 2024• 0 citations
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Abstract

Larger multi-qubit quantum gates allow shallower, more efficient quantum circuits, which could decrease the prohibitive effect of noise on algorithms for noisy intermediate-scale quantum (NISQ) devices and fault-tolerant error correction schemes. Such multi-qubit gates can potentially be generated by time-independent Hamiltonians comprising only physical (one- and two-local) interaction terms. Here, we present an algorithm that finds the time-independent Hamiltonian for a target quantum gate on $n$ qubits by using the geodesic on the Riemannian manifold of $\mathrm{SU}(2^n)$. Differential programming is used to determine how the Hamiltonian should be updated in order to follow the geodesic to the target unitary as closely as possible. We show that our geodesic algorithm outperforms gradient descent methods for standard multi-qubit gates such as Toffoli and Fredkin. The geodesic algorithm is then used to find previously unavailable multi-qubit gates implementing high fidelity parity checks, which could be used in a wide array of quantum codes and increase the clock speed of fault-tolerant quantum computers. The geodesic algorithm is demonstrated on an example relevant to current experimental hardware, illustrating a circuit speed up.

DOI
10.48550/arxiv.2401.05973

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Larger multi-qubit quantum gates allow shallower, more efficient quantum circuits, which could decrease the prohibitive effect of noise on algorithms for noisy intermediate-scale quantum (NISQ) devices and fault-tolerant error correction schemes. Such multi-qubit gates can potentially be generated by time-independent Hamiltonians comprising only physical (one- and two-local) interaction terms. Here, we present an algorithm that finds the time-independent Hamiltonian for a target quantum gate on nnn qubits by using the geodesic on the Riemannian manifold of SU(2n)\mathrm{SU}(2^n)SU(2n). Differential programming is used to determine how the Hamiltonian should be updated in order to follow the geodesic to the target unitary as closely as possible. We show that our geodesic algorithm outperforms gradient descent methods for standard multi-qubit gates such as Toffoli and Fredkin. The geodesic algorithm is then used to find previously unavailable multi-qubit gates implementing high fidelity parity checks, which could be used in a wide array of quantum codes and increase the clock speed of fault-tolerant quantum computers. The geodesic algorithm is demonstrated on an example relevant to current experimental hardware, illustrating a circuit speed up.
Larger multiqubit quantum gates allow shallower, more efficient quantum circuits, which could decrease the prohibitive effect of noise on algorithms for noisy intermediate-scale quantum (NISQ) devices and fault-tolerant error correction schemes. Such multiqubit gates can potentially be generated by time-independent Hamiltonians comprising only physical (one- and two-local) interaction terms. Here, we present an algorithm that finds the time-independent Hamiltonian for a target quantum gate on n qubits by using the geodesic on the Riemannian manifold of SU ( 2 n ) . Differential programming is used to determine how the Hamiltonian should be updated to follow the geodesic to the target unitary as closely as possible. We show that our geodesic algorithm outperforms gradient descent methods for standard multiqubit gates such as Toffoli and Fredkin. The geodesic algorithm is then used to find previously unavailable multiqubit gates implementing high-fidelity parity checks, which could be used in a wide array of quantum codes and increase the clock speed of fault-tolerant quantum computers. The geodesic algorithm is demonstrated on an example relevant to current experimental hardware, illustrating a circuit speed up.
Implementing quantum operations in the form of natural Hamiltonian dynamics is desirable, since they almost require no external control or feedback. In this work, we propose a NISQ-friendly quantum-classical hybrid approach to designing a time-independent Hamiltonian that generates a given multi-qubit unitary. In particular, we execute a Variational Quantum Algorithm, whose ansatz is carefully chosen to be a sequence of appropriately parametrized unitaries describing at most two-qubit nearest neighbour interactions, dictating the target unitary. Subsequently, we apply our approach to simulate multi-qubit target gates, with and without stochastic noise. We demonstrate that our strategy allows us to implement a Toffoli gate with sufficiently high fidelity, as compared to the other similar techniques. Our approach is an example of the usage of quantum computing for the design of quantum computers.
A unitary evolution in time may be treated as a curve in the manifold of the special unitary group. The length of such a curve can be related to the energetic cost of the associated computation, meaning a geodesic curve identifies an energetically optimal path. In this work, we employ sub-Riemannian geometry on the manifold of the unitary group to obtain optimally designed Hamiltonians for generating single-qubit gates in an environment with the presence of dephasing noise as well as a two-qubit gate under a time-constant crosstalk interaction. The resulting geodesic equation involves knowing the initial conditions of the parameters that cannot be obtained analytically. We then introduce a random sampling method combined with a minimization function and a cost function to find initial conditions that lead to optimal control fields. We also compare the optimized control fields obtained from the solutions of the geodesic equation with those extracted from the well-known Krotov method. Both approaches provide high-fidelity values for the desired quantum gate implementation, but the geodesic method has the advantage of minimizing the required energy to execute the same task. These findings bring insights for the design of more efficient fields in the arsenal of optimal control theory.
Quantum computing faces challenges such as noise, short coherence time, and limited qubit connections. These challenges worsen as quantum circuits become more complex. One major issue is the increasing depth of quantum circuits. This research proposes an optimization framework targeting depth and gate count reduction in quantum circuits, specifically for Noisy Intermediate-Scale Quantum (NISQ) devices. The proposed approach combines unitary merging of single-qubit gates, CNOT cancellation, gate commuting, and rotation gate rewriting strategies. Consecutive gates acting on the same qubit, such asRx(θ1)⋅Rx(θ2)R_{x}\left(\theta_{1}\right) \cdot R_{x}\left(\theta_{2}\right)Rx​(θ1​)⋅Rx​(θ2​), are algebraically merged into a single rotation, while pairs of redundant CNOT gates are eliminated based on gate identity relations. The technique is implemented using Qiskit and validated across five diverse circuits including complex, random, and multi-qubit configurations. Experimental results show an average depth reduction of 33.33% and gate count reduction of 32.14%, with runtime improvement of up to 25%. For instance, an input circuit with a depth of 7 and gate count of 11 was reduced to a depth of 2 and 4 gates. All optimized circuits preserve functional correctness with a fidelityF≥0.99F \geq 0.99F≥0.99. High-resolution circuit diagrams are presented to visually demonstrate improvements before and after optimization. Additionally, global phase shifts such aseiπ/4e^{i \pi / 4}eiπ/4are preserved or analytically characterized where relevant. This work enhances the viability of quantum computations on near-term hardware and opens pathways for future AI-driven quantum optimizations.
Fault-tolerant quantum computing requires encoding the quantum information into logical qubits and performing the quantum information processing in a code-space. Quantum error correction codes, then, can be employed to diagnose and remove the possible errors in the quantum information, thereby avoiding the loss of information. Although a series of single- and two-qubit gates can be employed to construct a quantum error correcting circuit, however this decomposition approach is not practically desirable because it leads to circuits with long operation times. An alternative approach to designing a fast quantum circuit is to design quantum gates that act on a multi-qubit gate. Here I devise quantum control schemes to design high-fidelity single-shot multi-qubit (up to three) quantum gates. Quantum control task is to steer quantum dynamics towards closely realizing specific quantum operation by varying the external control parameters (external field) such that the resultant evolution closely approximates the desired evolution. A set of instructions that determines the control parameters, and hence the effectiveness of the control scheme, is called a policy. Machine learning algorithms can be employed to find successful policies for designing quantum gates. In particular, we employ supervised machine learning techniques to generate these successful policies. Finding successful policies is a feasibility problem for which optimization algorithms can be employed. Greedy algorithms are at the heart of machine learning techniques. They converge faster onto a successful policy and require less-computational resource than non-greedy algorithms. However, there is no guarantee that greedy algorithms succeed to a feasible solution when there exist constraints on i) gate operation time ii) computational resources, and iii) experimental resources. Our results show the failure of standard greedy machine learning algorithms and the superiority of non-greedy machine learning algorithms over greedy ones for designing quantum logic gates, when there exist constraints on the quantum system. We have also observed the failure of existing greedy and non-greedy techniques for designing high-fidelity three-qubit gates. Hence, we devised our machine learning technique called Subspace-Selective Self-adaptive Differential Evolution (SuSSADE). Each three-qubit gate designed by SuSSADE operates as fast as an entangling two-qubit gate under the same experimental constraints.
Abstract The development of a universal fault-tolerant quantum computer that can solve efficiently various difficult computational problems is an outstanding challenge for science and technology. In this work, we propose a technique for an efficient implementation of quantum algorithms with multilevel quantum systems (qudits). Our method uses a transpilation of a circuit in the standard qubit form, which depends on the characteristics of a qudit-based processor, such as the number of available qudits and the number of accessible levels. This approach provides a qubit-to-qudit mapping and comparison to a standard realization of quantum algorithms highlighting potential advantages of qudits. We provide an explicit scheme of transpiling qubit circuits into sequences of single-qudit and two-qudit gates taken from a particular universal set. We then illustrate our method by considering an example of an efficient implementation of a 6-qubit quantum algorithm with qudits. In this particular example, we demonstrate how using qudits allows a decreasing amount of two-body interactions in the qubit circuit implementation. We expect that our findings are of relevance for ongoing experiments with noisy intermediate-scale quantum devices that operate with information carriers allowing qudit encodings, such as trapped ions and neutral atoms, as well as optical and solid-state systems.