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.
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.
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.
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 1/8 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 1/9. Our second scheme provides a fully state independent realization reducing the number of non-local gates to O(2r1+2r2) as compared to the existing bound of O(2r1+r2) where r1 and r2 are the number of qubits to be removed as control in the qudits. The success probability of the realization is 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 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.
Unitary decomposition is a widely used method to map quantum algorithms to an arbitrary set of quantum gates. Efficient implementation of this decomposition allows for the translation of bigger unitary gates into elementary quantum operations, which is key to executing these algorithms on existing quantum computers. The decomposition can be used as an aggressive optimization method for the whole circuit, as well as to test part of an algorithm on a quantum accelerator. For the selection and implementation of the decomposition algorithm, perfect qubits are assumed. We base our decomposition technique on Quantum Shannon Decomposition, which generates O(344n) controlled-not gates for an n-qubit input gate. In addition, we implement optimizations to take advantage of the potential underlying structure in the input or intermediate matrices, as well as to minimize the execution time of the decomposition. Comparing our implementation to Qubiter and the UniversalQCompiler (UQC), we show that our implementation generates circuits that are much shorter than those of Qubiter and not much longer than the UQC. At the same time, it is also up to 10 times as fast as Qubiter and about 500 times as fast as the UQC.
The paradigm behind digital quantum computing inherits the idea of using binary information processing. Nature in fact gives much more rich structures of physical objects that can be used for encoding information, which is especially interesting in the quantum-mechanical domain. In this Colloquium several ideas are reviewed that indicate how multilevel quantum systems, also known as qudits, can be used for efficient realization of quantum algorithms, which are represented via standard qubit circuits. The focus in the Colloquium is on techniques for leveraging qudits for simplifying decomposition of multiqubit gates and for compressing quantum information by encoding multiple qubits in a single qudit. As discussed in the Colloquium, these approaches can be efficiently combined. This allows a reduction in the number of entangling (two-body) operations and the number of quantum information carriers used compared to straightforward qubit realizations. These theoretical schemes can be implemented with quantum computing platforms of various natures, such as trapped ions, neutral atoms, superconducting junctions, quantum light, spin systems, and molecules. The Colloquium concludes by summarizing a set of open problems whose resolution will be an important further step toward employing universal qudit-based processors for running qubit algorithms.
qiskit-qudits is a Qiskit extension that simulates d-level qudits by encoding each one into m = ceil(log_2 d) qubits. Qudit gates are exposed as ordinary Qiskit Gate and ControlledGate subclasses, and operations that are not unitary gates (measurement, reset, barrier, state preparation) as dedicated Instruction subclasses dispatched through a dedicated apply() hook; every gate carries both a dense encoded unitary (via NumPy's array protocol) and a qubit-level definition. Because d need not be a power of two, the encoded Hilbert space is generally larger than the logical one; the library resolves this by the identity-padding convention, in which every gate acts as the identity on the unphysical part of the encoded space. When every operand dimension is a power of two, gates decompose into a fixed, transpiler-recognisable cascade of standard qubit gates; otherwise the library falls back to exact dense unitary synthesis, so that dimensions 2 <= d <= 16 are supported exactly, not only powers of two. This paper describes the software's circuit model, gate hierarchy, decomposition strategy, and measurement/decoding machinery, states its limitations, and verifies the implementation numerically: the qudit QFT against the discrete Fourier transform for both power-of-two and non-power-of-two d, extending the check of the underlying theory paper, which could only be run for d = 2^m, and every gate's emitted decomposition against its dense unitary across the whole gate set.