No file available
To read the file of this research,
you can request a copy directly from the authors.
Mixed-Dimensional Qudit State Preparation Using Edge-Weighted Decision Diagrams
Preprints and early-stage research may not have been peer reviewed yet.
Abstract
Quantum computers have the potential to solve important problems which are fundamentally intractable on a classical computer. The underlying physics of quantum computing platforms supports using multi-valued logic, which promises a boost in performance over the prevailing two-level logic. One key element to exploiting this potential is the capability to efficiently prepare quantum states for multi-valued, or qudit, systems. Due to the time sensitivity of quantum computers, the circuits to prepare the required states have to be as short as possible. In this paper, we investigate quantum state preparation with a focus on mixed-dimensional systems, where the individual qudits may have different dimensionalities. The proposed approach automatically realizes quantum circuits constructing a corresponding mixed-dimensional quantum state. To this end, decision diagrams are used as a compact representation of the quantum state to be realized. We further incorporate the ability to approximate the quantum state to enable a finely controlled trade-off between accuracy, memory complexity, and number of operations in the circuit. Empirical evaluations demonstrate the effectiveness of the proposed approach in facilitating fast and scalable quantum state preparation, with performance directly linked to the size of the decision diagram. The implementation is freely available as part of Munich Quantum Toolkit~(MQT), under the framework MQT Qudits at github.com/cda-tum/mqt-qudits.
ResearchGate has not been able to resolve any citations for this publication.
We characterize control of a qutrit implemented in the lowest three energy levels of a capacitively shunted flux-biased superconducting circuit. Randomized benchmarking over the qutrit Clifford group yields an average fidelity of 98.89±0.05%. For a selected subset of the Clifford group, we perform quantum process tomography and observe the behavior of repeated gate sequences. Each qutrit gate is generated using only two-state rotations via a method applicable to any unitary. We find that errors are due primarily to decoherence and have a significant contribution from level shifts. This work demonstrates high-fidelity qutrit control and outlines avenues for future work on the optimal control of superconducting qudits.
Most quantum computing architectures to date natively supportmulti-malued logic, albeit being typically operated in a binary fash-ion. Multi-valued, or qudit, quantum processors have access tomuch richer forms of quantum entanglement, which promise tosignificantly boost the performance and usefulness of quantum de-vices. However, much of the theory as well as corresponding designmethods required for exploiting such hardware remain insufficientand generalizations from qubits are not straightforward. A partic-ular challenge is the compilation of quantum circuits into sets ofnative qudit gates supported by state-of-the-art quantum hardware.In this work, we address this challenge by introducing a completeworkflow for compiling any two-qudit unitary into an arbitrarynative gate set. Case studies demonstrate the feasibility of both, theproposed approach as well as the corresponding implementation(which is freely available at github.com/cda-tum/qudit-entanglement-compilation).
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 https://github.com/cda-tum/qudit-compilation under an open-source license.
Most quantum computers use binary encoding to store information in qubits—the quantum analogue of classical bits. Yet, the underlying physical hardware consists of information carriers that are not necessarily binary, but typically exhibit a rich multilevel structure. Operating them as qubits artificially restricts their degrees of freedom to two energy levels¹. Meanwhile, a wide range of applications—from quantum chemistry² to quantum simulation³—would benefit from access to higher-dimensional Hilbert spaces, which qubit-based quantum computers can only emulate⁴. Here we demonstrate a universal quantum processor using trapped ions that act as qudits with a local Hilbert-space dimension of up to seven. With a performance similar to qubit quantum processors⁵, this approach enables the native simulation of high-dimensional quantum systems³, as well as more efficient implementation of qubit-based algorithms6,7.
Quantum computers promise to solve important problems faster than conventional computers ever could. Underneath is a fundamentally different computational primitive that introduces new challenges for the development of software tools that aid designers of corresponding quantum algorithms. The different computational primitives render classical simulation of quantum circuits particularly challenging. While the logic simulation of conventional circuits is comparatively simple with linear complexity with respect to the number of gates, quantum circuit simulation has to deal with the exponential memory requirements to represent quantum states on non-quantum hardware with respect to the number of qubits. Decision Diagrams (DDs) address this challenge through exploitation of redundancies in matrices and vectors to provide significantly more compact representations in many cases. Moreover, the probabilistic nature of quantum computations enables another angle to tackle the complexity: Quantum algorithms are resistant to some degree against small inaccuracies in the quantum state as these only lead to small changes in the outcome probabilities. We propose to exploit this resistance against (small) errors to gain even more compact decision diagrams.
In this work, we investigate the potential of approximation in quantum circuit simulation in detail. To this end, we first present four dedicated schemes that exploit the error resistance and efficiently approximate quantum states represented by decision diagrams. Subsequently, we propose two simulation strategies that utilize those approximations schemes in order to improve the efficiency of DD-based quantum circuit simulation, while, at the same time, allowing the user to control the resulting degradation in accuracy. We empirically show that the proposed approximation schemes reduce the size of decision diagrams substantially and also analytically prove the effect of multiple approximations on the attained accuracy. Eventually, this enables speed-ups of the resulting approximate quantum circuit simulation of up to several orders of magnitudes—again, while controlling the fidelity of the result.
In this paper, we propose a quantum algorithm for recommendation systems which incorporates the contextual information of users to the personalized recommendation. The preference information of users is encoded in a third-order tensor of dimension N which can be approximated by the truncated tensor singular value decomposition (t-svd) of the subsample tensor. Unlike the classical algorithm that reconstructs the approximated preference tensor using truncated t-svd, our quantum algorithm obtains the recommended product under certain context by measuring the output quantum state corresponding to an approximation of a user’s dynamic preferences. The algorithm achieves the time complexity \(\mathcal {O}(\sqrt{k}N\mathrm{polylog}(N))\), compared to the classical counterpart with complexity \(\mathcal {O}(kN^3)\), where k is the truncated tubal rank.
Qudit is a multi-level computational unit alternative to the conventional 2-level qubit. Compared to qubit, qudit provides a larger state space to store and process information, and thus can provide reduction of the circuit complexity, simplification of the experimental setup and enhancement of the algorithm efficiency. This review provides an overview of qudit-based quantum computing covering a variety of topics ranging from circuit building, algorithm design, to experimental methods. We first discuss the qudit gate universality and a variety of qudit gates including the pi/8 gate, the SWAP gate, and the multi-level controlled-gate. We then present the qudit version of several representative quantum algorithms including the Deutsch-Jozsa algorithm, the quantum Fourier transform, and the phase estimation algorithm. Finally we discuss various physical realizations for qudit computation such as the photonic platform, iron trap, and nuclear magnetic resonance.
Quantum computing promises substantial speedups by exploiting quantum mechanical phenomena such as superposition and entanglement. Corresponding design methods require efficient means of representation and manipulation of quantum functionality.
In the classical domain, decision diagrams have been successfully employed as a powerful alternative to straightforward means such as truth tables.
This motivated extensive research on whether decision diagrams provide similar potential in the quantum domain---resulting in new types of decision diagrams capable of substantially reducing the complexity of representing quantum states and functionality.
From an implementation perspective, many concepts and techniques from the classical domain can be re-used in order to implement decision diagrams packages for the quantum realm. However, new problems---namely how to efficiently handle complex numbers---arise.
In this work, we propose a solution to overcome these problems.
Experimental evaluations confirm that this yields improvements of orders of magnitude in the runtime needed to create and to utilize these decision diagrams.
The resulting implementation is publicly available as a quantum DD package at http://iic.jku.at/eda/research/quantum_dd.
We present the problem of approximating the time-evolution operator e−iH^t to error ϵ , where the Hamiltonian H^=(⟨G|⊗I^)U^(|G⟩⊗I^) is the projection of a unitary oracle U^ onto the state |G⟩ created by another unitary oracle. Our algorithm solves this with a query complexity O(t+log(1/ϵ)) to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are d -sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where H^ is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any H^ in an invariant SU(2) subspace. A large class of operator functions of H^ can then be computed with optimal query complexity, of which e−iH^t is a special case.
Quantum superdense coding protocols enhance channel capacity by using shared quantum entanglement between two users. The channel capacity can be as high as 2 when one uses entangled qubits. However, this limit can be surpassed by using high-dimensional entanglement. We report an experiment that exceeds the limit using high-quality entangled ququarts with fidelities up to 0.98, demonstrating a channel capacity of 2.09 ± 0.01. The measured channel capacity is also higher than that obtained when transmitting only one ququart. We use the setup to transmit a five-color image with a fidelity of 0.952. Our experiment shows the great advantage of high-dimensional entanglement and will stimulate research on high-dimensional quantum information processes.
With quantum computers of significant size now on the horizon, we should understand how to best exploit their initially limited abilities. To this end, we aim to identify a practical problem that is beyond the reach of current classical computers, but that requires the fewest resources for a quantum computer. We consider quantum simulation of spin systems, which could be applied to understand condensed matter phenomena. We synthesize explicit circuits for three leading quantum simulation algorithms, employing diverse techniques to tighten error bounds and optimize circuit implementations. Quantum signal processing appears to be preferred among algorithms with rigorous performance guarantees, whereas higher-order product formulas prevail if empirical error estimates suffice. Our circuits are orders of magnitude smaller than those for the simplest classically-infeasible instances of factoring and quantum chemistry.
Quantum mechanics predicts a number of at first sight counterintuitive
phenomena. It is therefore a question whether our intuition is the best way to
find new experiments. Here we report the development of the computer algorithm
Melvin which is able to find new experimental implementations for the creation
and manipulation of complex quantum states. And indeed, the discovered
experiments extensively use unfamiliar and asymmetric techniques which are
challenging to understand intuitively. The results range from the first
implementation of a high-dimensional Greenberger-Horne-Zeilinger (GHZ) state,
to a vast variety of experiments for asymmetrically entangled quantum states -
a feature that can only exist when both the number of involved parties and
dimensions is larger than 2. Additionally, new types of high-dimensional
transformations are found that perform cyclic operations. Melvin autonomously
learns from solutions for simpler systems, which significantly speeds up the
discovery rate of more complex experiments. The ability to automate the design
of a quantum experiment can be applied to many quantum systems and allows the
physical realization of quantum states previously thought of only on paper.
We propose the generalized controlled X (GCX) gate as the two-qudit
elementary gate, and based on Cartan decomposition, we also give the one-qudit
elementary gates. Then we discuss the physical implementation of these
elementary gates and show that it is feasible with current technology. With
these elementary gates many important qudit quantum gates can be synthesized
conveniently. We provide efficient methods for the synthesis of various kinds
of controlled qudit gates and greatly simplify the synthesis of existing
generic multi-valued quantum circuits. Moreover, we generalize the quantum
Shannon decomposition (QSD), the most powerful technique for the synthesis of
generic qubit circuits, to the qudit case. A comparison of ququart (d=4)
circuits and qubit circuits reveals that using ququart circuits may have an
advantage over the qubit circuits in the synthesis of quantum circuits.
Quantum computation offers the potential to solve fundamental yet otherwise
intractable problems across a range of active fields of research. Recently,
universal quantum-logic gate sets - the building blocks for a quantum computer
- have been demonstrated in several physical architectures. A serious obstacle
to a full-scale implementation is the sheer number of these gates required to
implement even small quantum algorithms. Here we present and demonstrate a
general technique that harnesses higher dimensions of quantum systems to
significantly reduce this number, allowing the construction of key quantum
circuits with existing technology. We are thereby able to present the first
implementation of two key quantum circuits: the three-qubit Toffoli and the
two-qubit controlled-unitary. The gates are realised in a linear optical
architecture, which would otherwise be absolutely infeasible with current
technology.
A frequent starting point of quantum computation platforms is the two-state quantum system, i.e., the qubit. However, in the context of integer optimization problems, relevant to scheduling optimization and operations research, it is often more resource-efficient to employ quantum systems with more than two basis states, so-called qudits. Here, we discuss the quantum approximate optimization algorithm (QAOA) for qudit systems. We illustrate how the QAOA can be used to formulate a variety of integer optimization problems such as graph coloring problems or electric vehicle charging optimization. In addition, we comment on the implementation of constraints and describe three methods to include these in a quantum circuit of a QAOA by penalty contributions to the cost Hamiltonian, conditional gates using ancilla qubits, and a dynamical decoupling strategy. Finally, as a showcase of qudit-based QAOA, we present numerical results for a charging optimization problem mapped onto a maximum-k-graph-coloring problem. Our work illustrates the flexibility of qudit systems to solve integer optimization problems.
Quantum state preparation is an important subroutine for quantum computing. We show that any n-qubit quantum state can be prepared with a Θ(n)-depth circuit using only single- and two-qubit gates, although with a cost of an exponential amount of ancillary qubits. On the other hand, for sparse quantum states with d⩾2 nonzero entries, we can reduce the circuit depth to Θ(log(nd)) with O(ndlogd) ancillary qubits. The algorithm for sparse states is exponentially faster than best-known results and the number of ancillary qubits is nearly optimal and only increases polynomially with the system size. We discuss applications of the results in different quantum computing tasks, such as Hamiltonian simulation, solving linear systems of equations, and realizing quantum random access memories, and find cases with exponential reductions of the circuit depth for all these three tasks. In particular, using our algorithm, we find a family of linear system solving problems enjoying exponential speedups, even compared to the best-known quantum and classical dequantization algorithms.
Loading classical data into quantum registers is one of the most important primitives of quantum computing. While the complexity of preparing a generic quantum state is exponential in the number of qubits, in many practical tasks the state to prepare has a certain structure that allows for faster preparation. In this paper, we consider quantum states that can be efficiently represented by (reduced) decision diagrams, a versatile data structure for the representation and analysis of Boolean functions. We design an algorithm that utilizes the structure of decision diagrams to prepare their associated quantum states. Our algorithm has a circuit complexity that is linear in the number of paths in the decision diagram. Numerical experiments show that our algorithm reduces the circuit complexity by up to 31.85% compared to the state-of-the-art algorithm, when preparing generic n-qubit states with n3 nonzero amplitudes. Additionally, for states with sparse decision diagrams, including the initial state of the quantum Byzantine agreement protocol, our algorithm reduces the number of controlled-nots by 86.61–99.9%.
We present a gauge invariant digitization of (1+1)d scalar quantum electrodynamics for an arbitrary spin truncation for qudit-based quantum computers. We provide a construction of the Trotter operator in terms of a universal qudit-gate set. The cost savings of using a qutrit based spin-1 encoding versus a qubit encoding are illustrated. We show that a simple initial state could be simulated on current qutrit based hardware using noisy simulations for two different native gate set.
Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chemistry communities over the past century. Although many approximation methods have been introduced, the complexity of quantum mechanics remains hard to appease. The advent of quantum computation brings new pathways to navigate this challenging and complex landscape. By manipulating quantum states of matter and taking advantage of their unique features such as superposition and entanglement, quantum computers promise to efficiently deliver accurate results for many important problems in quantum chemistry, such as the electronic structure of molecules. In the past two decades, significant advances have been made in developing algorithms and physical hardware for quantum computing, heralding a revolution in simulation of quantum systems. This Review provides an overview of the algorithms and results that are relevant for quantum chemistry. The intended audience is both quantum chemists who seek to learn more about quantum computing and quantum computing researchers who would like to explore applications in quantum chemistry.
Quantum mechanical phenomena such as phase shifts, superposition, and entanglement show promise in use for computation. Suitable technologies for the modeling and design of quantum computers and other information processing techniques that exploit quantum mechanical principles are in the range of vision. Quantum algorithms that significantly speed up the process of solving several important computation problems have been proposed in the past. The most common representation of quantum mechanical phenomena are transformation matrices. However, the transformation matrices grow exponentially with the size of a quantum system and, thus, pose significant challenges for efficient representation and manipulation of quantum functionality. In order to address this problem, first approaches for the representation of quantum systems in terms of decision diagrams have been proposed. One very promising approach is given by Quantum Multiple-Valued Decision Diagrams (QMDDs) which are able to efficiently represent transformation matrices and also inherently support multiple-valued basis states offered by many physical quantum systems. However, the initial proposal of QMDDs was lacking in a formal basis and did not allow, e.g., the change of the variable order-an established core functionality in decision diagrams which is crucial for determining more compact representations. Because of this, the full potential of QMDDs or decision diagrams for quantum functionality in general has not been fully exploited yet. In this paper, we present a refined definition of QMDDs for the general quantum case. Furthermore, we provide significantly improved computational methods for their use and manipulation and show that the resulting representation satisfies important criteria for a decision diagram, i.e., compactness and canonicity. An experimental evaluation confirms the efficiency of QMDDs.
We analyze the efficiency of quantum simulations of fermionic and bosonic
models in trapped ions. In particular, we study the optimal time of entangling
gates and the required number of total elementary gates. Furthermore, we
exemplify these estimations in the light of quantum simulations of quantum
field theories, condensed-matter physics, and quantum chemistry. Finally, we
show that trapped-ion technologies are a suitable platform for implementing
quantum simulations involving interacting fermionic and bosonic modes, paving
the way for overcoming classical computers in the near future.
The usual way to reveal properties of an unknown quantum state, given many
copies of a system in that state, is to perform measurements of different
observables and to analyze the measurement results statistically. Here we show
that the unknown quantum state can play an active role in its own analysis. In
particular, given multiple copies of a quantum system with density matrix \rho,
then it is possible to perform the unitary transformation e^{-i\rho t}. As a
result, one can create quantum coherence among different copies of the system
to perform quantum principal component analysis, revealing the eigenvectors
corresponding to the large eigenvalues of the unknown state in time
exponentially faster than any existing algorithm.
We present a polynomial quantum algorithm for the Abelian stabilizer problem which includes both factoring and the discrete logarithm. Thus we extend famous Shor's results [7]. Our method is based on a procedure for measuring an eigenvalue of a unitary operator. Another application of this procedure is a polynomial quantum Fourier transform algorithm for an arbitrary finite Abelian group. The paper also contains a rather detailed introduction to the theory of quantum computation.
Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b(-->), find a vector x(-->) such that Ax(-->) = b(-->). We consider the case where one does not need to know the solution x(-->) itself, but rather an approximation of the expectation value of some operator associated with x(-->), e.g., x(-->)(dagger) Mx(-->) for some matrix M. In this case, when A is sparse, N x N and has condition number kappa, the fastest known classical algorithms can find x(-->) and estimate x(-->)(dagger) Mx(-->) in time scaling roughly as N square root(kappa). Here, we exhibit a quantum algorithm for estimating x(-->)(dagger) Mx(-->) whose runtime is a polynomial of log(N) and kappa. Indeed, for small values of kappa [i.e., poly log(N)], we prove (using some common complexity-theoretic assumptions) that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm.
In this paper, we present a novel structure, QuantumMultiple- valued Decision Diagrams (QMDD), specifically designed to represent and manipulate the matrices encountered in the specification of reversible and quantum gates and circuits, both binary and multiple-valued. QMDD use many common decision diagram techniques, ideas introduced in QuIDDPro and novel techniques introduced here. Building the QMDD for the matrices for individual gates and the subsequent construction of the QMDD for the matrix describing a circuit are discussed. A prototype C implementation is described and experimental results are given that show the new structure is a promising and compact representation for reversible and quantum logic circuits.
A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored. AMS subject classifications: 82P10, 11Y05, 68Q10. 1 Introduction One of the first results in the mathematics of computation, which underlies the subsequent development of much of theoretical computer science, was the distinction between compu...
Imagine a phone directory containing N names arranged in completely random order. In order to find someone's phone number with a 50% probability, any classical algorithm (whether deterministic or probabilistic) will need to look at a minimum of N/2 names. Quantum mechanical systems can be in a superposition of states and simultaneously examine multiple names. By properly adjusting the phases of various operations, successful computations reinforce each other while others interfere randomly. As a result, the desired phone number can be obtained in only O(sqrt(N)) steps. The algorithm is within a small constant factor of the fastest possible quantum mechanical algorithm.
A proof of Bell's theorem without inequalities valid for both inequivalent classes of three-qubit entangled states under local operations assisted by classical communication, namely Greenberger-Horne-Zeilinger (GHZ) and W, is described. This proof leads to a Bell inequality that allows more conclusive tests of Bell's theorem for three-qubit systems. Another Bell inequality involving both tri- and bipartite correlations is introduced which illustrates the different violations of local realism exhibited by the GHZ and W states. Comment: REVTeX4, 5 pages, 3 figures