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IEEE International Conference on Quantum Software (QSW)

Compression of Qubit Circuits: Mapping to Mixed-Dimensional Quantum Systems

Kevin Mato, Stefan Hillmich, Robert Wille

Published Jul 1, 2023• 7 citations
Publisher

Abstract

Quantum computers are becoming a reality thanks to the accomplishments made in recent years. The quantum computers available today offer hundreds of qubits but are still limited in the number of operations they can perform before errors accumulate and the quantum state decays. In regard to the error accumulation, non-local operations such as CX or CZ are main contributors. One promising solution to reduce the number of required non-local operations is to make a more efficient use of the quantum hardware by exploiting the inherent high-dimensional capabilities of quantum systems. In a process called circuit compression, non-local operations between qubits are mapped to local operations in qudits, i.e., higher-dimensional systems. In this work, we present a strategy for enabling quantum circuit compression with the aim of mapping clusters of qubits in a given circuit to the mixed-dimensional qudits of the target hardware. Further, we discuss the principles of circuit compression as well as the physical structure of qubits and qudits, before introducing a new representation that captures the essence of quantum operations, affecting the different logical levels in the quantum states in nodes and edges of a graph. Based on this, we propose an automated approach for mapping qubit circuits of arbitrary gate sets to mixed-dimensional quantum systems, lowering the number of non-local operations. Empirical evaluations confirm the effectiveness of the proposed approach reducing the number of non-local operations by up to 50% for almost half of the cases. Finally, the corresponding source code is available freely at github.com/cda-tdum/qudit-compression.

DOI
10.1109/qsw59989.2023.00027

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Quantum computers are becoming a reality thanks to the accomplishments made in recent years. The quantum computers available today offer hundreds of qubits but are still limited in the number of operations they can perform before errors accumulate and the quantum state decays. In regard to the error accumulation, non-local operations such as CX or CZ are main contributors. One promising solution to reduce the number of required non-local operations is to make a more efficient use of the quantum hardware by exploiting the inherent high-dimensional capabilities of quantum systems. In a process called circuit compression, non-local operations between qubits are mapped to local operations in qudits, i.e., higher-dimensional systems. In this work, we present a strategy for enabling quantum circuit compression with the aim of mapping clusters of qubits in a given circuit to the mixed-dimensional qudits of the target hardware. Further, we discuss the principles of circuit compression as well as the physical structure of qubits and qudits, before introducing a new representation that captures the essence of quantum operations, affecting the different logical levels in the quantum states in nodes and edges of a graph. Based on this, we propose an automated approach for mapping qubit circuits of arbitrary gate sets to mixed-dimensional quantum systems, lowering the number of non-local operations. Empirical evaluations confirm the effectiveness of the proposed approach reducing the number of non-local operations by up to 50% for almost half of the cases. Finally, the corresponding source code is available freely at github.com/cda-tdum/qudit-compression.
Quantum computers promise to solve several categories of problems faster than classical computers ever could. Current research mostly focuses on qubits, i.e., systems where the unit of information can assume only two levels. However, the underlying physics of most (if not all) of the technological platforms supports more than two levels, commonly referred to as qudits. Performing computations with qudits increases the overall complexity while, at the same time, reducing the number of operations and providing a lower error rate. Furthermore, qudits with different number of levels can be mixed in one system to ease the experimental control and keep representations as compact as possible. Exploiting these capabilities requires dedicated software support to tackle the increased complexity in an automated and efficient fashion. In this paper, we present a qudit simulator that handles mixed-dimensional systems based on Decision Diagrams (DDs). More precisely, we discuss the type of decision diagram introduced as underlying data structure as well as the resulting implementation. Experimental evaluations demonstrate that the proposed solution is capable of efficiently simulating mixed-dimensional quantum circuits, with specific use cases including more than 100 qudits in one circuit. The source code of the simulator is available via github.com/cda-tdum/MiSiM under the MIT license.
Gate-based universal quantum computation is formulated in terms of two types of operations: local single-qubit gates, which are typically easily implementable, and two-qubit entangling gates, whose faithful implementation remains one of the major experimental challenges since it requires controlled interactions between individual systems. To make the most of quantum hardware it is crucial to process information in the most efficient way. One promising avenue is to use higher-dimensional systems, qudits, as the fundamental units of quantum information, in order to replace a fraction of the qubit-entangling gates with qudit-local gates. Here, we show how the complexity of multi-qubit circuits can be lowered significantly by employing qudit encodings, which we quantify by considering exemplary circuits with exactly known (multi-qubit) gate complexity. We discuss general principles for circuit compression, derive upper and lower bounds on the achievable advantage, and highlight the key role played by entanglement and the available gate set. Explicit experimental schemes for photonic as well as for trapped-ion implementations are provided and demonstrate a significant expected gain in circuit performance for both platforms.
Most quantum computing architectures to date natively support multi-valued logic, albeit being typically operated in a binary fashion. Multi-valued, or qudit, quantum processors have access to much richer forms of quantum entanglement, which promise to significantly boost the performance and usefulness of quantum devices. However, much of the theory as well as corresponding design methods required for exploiting such hardware remain insufficient and generalizations from qubits are not straightforward. A particular challenge is the compilation of quantum circuits into sets of native qudit gates supported by state-of-the-art quantum hardware. In this work, we address this challenge by introducing a complete workflow for compiling any two-qudit unitary into an arbitrary native gate set. Case studies demonstrate the feasibility of both, the proposed approach as well as the corresponding implementation (which is freely available at github.com/cda-tum/qudit-entanglement-compilation).
Linear optical quantum computation (LOQC) offers a promising platform for scalable quantum information processing, but its scalability is fundamentally constrained by the probabilistic nature of non-local entangling gates. Qudit circuit compression schemes mitigate this issue by encoding multiple qubits onto qudits. However, these schemes become inefficient when only a subset of the encoded qubits is required to participate in the non-local entangling gate, leading to an exponential increase in the number of non-local gates. In this Letter, we address this bottleneck by demonstrating the existence of multi-level control-Z (CZ) gates for qudits encoded in multiple spatial modes in LOQC. Unlike conventional two-level CZ gates, which act only on a single pair of modes, multi-level CZ gates impart a conditional phase shift for an arbitrarily chosen subset of the spatial modes. We present two explicit linear optical schemes that realize such operations, illustrating a fundamental trade-off between prior information about the input quantum state and the physical resources required. The first scheme is realized with a constant success probability of 1/81/81/8 independent of the qudit dimension using a single non-local entangling gate, at the cost of state dependence, which is significantly better than the current success probability of 1/91/91/9. Our second scheme provides a fully state independent realization reducing the number of non-local gates to O(2r1+2r2)\mathcal{O}(2^{r_1}+2^{r_2})O(2r1​+2r2​) as compared to the existing bound of O(2r1+r2)\mathcal{O}(2^{r_1+r_2})O(2r1​+r2​) where r1r_1r1​ and r2r_2r2​ are the number of qubits to be removed as control in the qudits. The success probability of the realization is 12(18)2r1+2r2\frac{1}{2} \left(\frac{1}{8}\right)^{2^{r_1}+2^{r_2}}21​(81​)2r1​+2r2​. When combined with qudit circuit compression schemes, our results improve upon a key scalability limitation and significantly improve the efficiency of LOQC architectures.
Quantum computers have the potential to solve some important industrial and scientific problems with greater efficiency than classical computers. While most current realizations focus on two-level qubits, the underlying physics used in most hardware is capable of extending the concepts to a multi-level logic—enabling the use of qudits, which promise higher computational power and lower error rates. Based on a strong theoretical backing and motivated by recent physical accomplishments, this also calls for methods and tools for compiling quantum circuits to those devices. To enable efficient qudit compilation, we introduce the concept of an energy coupling graph for single-qudit systems and provide an adaptive algorithm that leverages this representation for compiling arbitrary unitaries. This leads to significant improvements over the state-of-the-art compilation scheme and, additionally, provides an option to trade-off worst-case costs and run-time. The developed compiler is available via github.com/cda-tum/qudit-compilation under an open-source license.
We present qubit logic on qudits (QLOQ), a compression scheme in which the qubits from a hardware agnostic circuit are divided into groups of various sizes, and each group is mapped to a physical qudit for computation. QLOQ circuits have qubit-logic inputs, outputs, and gates, making them compatible with existing qubit-based algorithms and Hamiltonians. We show that arbitrary qubit-logic unitaries can in principle be implemented with significantly fewer two-level (qubit) physical entangling gates in QLOQ than in qubit encoding. We achieve this advantage in practice for two applications: variational quantum algorithms, and unitary decomposition. The variational quantum eigensolver (VQE) for LiH took 5 hours using QLOQ on one of Quandela's cloud-accessible photonic quantum computers, whereas it would have taken 4.39 years in qubit encoding. We also provide a QLOQ version of the Quantum Shannon Decomposition, which not only outperforms previous qudit-based proposals, but also beats the theoretical lower bound on the CNOT cost of unitary decomposition in qubit encoding.