Continuous systems are studied in many branches of modern physics, such as high-energy physics, cosmology, condensed matter physics, quantum chemistry, and field theories. Such systems are expected to benefit from the substantial advantages in computational power of quantum computers. The continuous-variable paradigm of quantum computation provides the most natural computational formalism for these tasks. However, most existing quantum hardware is based on discrete-variable systems. We address this fundamental discrepancy by providing a rigorous framework for translating native continuous-variable algorithms onto qubit-based quantum processors. This mapping is constructed from a gate-based model of continuous-variable quantum computers, consisting of states and operations built from a polynomial sequence of elementary gates in a finite set, with total energy polynomial in the number of modes. We prove that, under realistic constraints, a gate-based model of continuous-variable quantum computers can be efficiently simulated using discrete-variable devices, thereby establishing a computational equivalence between these paradigms.
Continuous systems are studied in many branches of modern physics, such as high-energy physics, cosmology, condensed matter physics, quantum chemistry, and field theories. Such systems are expected to benefit from the substantial advantages in computational power of quantum computers. The continuous-variable paradigm of quantum computation provides the most natural computational formalism for these tasks. However, most existing quantum hardware is based on discrete-variable systems. We address this fundamental discrepancy by providing a rigorous framework for translating native continuous-variable algorithms onto qubit-based quantum processors. This mapping is constructed from a gate-based model of continuous-variable quantum computers, consisting of states and operations built from a polynomial sequence of elementary gates in a finite set, with total energy polynomial in the number of modes. We prove that, under realistic constraints, a gate-based model of continuous-variable quantum computers can be efficiently simulated using discrete-variable devices, thereby establishing a computational equivalence between these paradigms.
This thesis investigates two complementary computational approaches to obtaining the outcome statistics of bosonic quantum experiments without directly performing the experiments themselves. In particular, we consider how continuous-variable quantum computations can be approximated using discrete-variable quantum systems and how bosonic outcome probabilities can be estimated using classical computational methods.The first part of the thesis focuses on the relation between continuous-variable (CV) and discrete-variable (DV) quantum computation. A framework is presented for approximating realistic CV quantum computations, consisting of polynomially bounded-energy states and a polynomial number of elementary gates, using DV quantum hardware. The proposed approximation error scales polynomially with the number of modes and can be made arbitrarily small by increasing the dimension of the simulating qudits. However, some technical aspects of the derivation remain to be rigorously justified. If these can be resolved, the result would establish a computational equivalence between the two paradigms. The remaining potential issues and possible approaches to resolving them are discussed.In the second part, we develop a framework for classically estimating individual outcome probabilities of general bosonic experiments using Monte Carlo methods based on (s)-ordered phase-space quasi-probability distributions. The resulting sampling complexity depends on properties of the phase-space quasi-probability distributions, in particular, their negativity and range, showing that negativity alone does not prevent efficient classical estimation. The outcome probability can be decomposed in several ways, giving rise to distinct trade-offs between these quantities. We discuss how experimentally relevant error channels can be incorporated into the framework and how they affect the classical estimation complexity, using pure loss as an illustrative example.Together, these results contribute to understanding the resources required to reproduce the outcome statistics of bosonic quantum experiments using both DV quantum and classical computation.
We provide a formal framework for Continuous-Variable Quantum Computing (CQC). While CQC is supported by photonic quantum hardware, we are not aware of a formal semantics for continuous-variable quantum programs nor of a unary Hoare logic for their verification. There are several technical obstacles to extending to CQC any of the formal frameworks available for Discrete-Variable Quantum Computing (DQC). Most importantly, continuous-variable quantum programs act on {\em infinite-dimensional} Hilbert spaces; their measurement outcomes are often {\em unbounded} and have expected values that are defined by an improper integral (or an infinite series), which may not converge. We overcome these challenges to give a formal semantics to a universal programming language for CQC and to provide the first Hoare logic for CQC. The assertions of our logic are built from polynomials over canonical observables. Besides proving relative completeness, we implement a symbolic weakest-precondition calculator for CQC based on our logic. Our tool has successfully verified CQC algorithms from textbooks and calculated their approximation errors for physically realizable implementations, proved the correctness (i.e., equivalence) of gate decompositions for CQC hardware, and computed the resource requirements (i.e., number of photon-number states) for achieving a desired accuracy in the classical simulation of continuous-variable quantum programs.
This tutorial offers a pedagogical guide to hybrid quantum processors that integrate discrete-variable (DV) qubits and continuous-variable (CV) oscillators. Aimed at computer scientists, engineers, and physicists, it provides an overview of the experimental, algorithmic, and architectural aspects of this novel and rapidly developing hardware model. Experimental realizations of this model include superconducting, trapped-ion, and neutral-atom platforms. By combining DV and CV components, hybrid oscillator-qubit processors enable a powerful new paradigm that offers complementary strengths for quantum control, error correction, computation, and simulation. Working toward the goal of a full-stack system connecting applications to CV-DV hardware, we define and formulate abstract machine models and instruction set architectures. These essential abstractions enable codesign of hardware and software, and resource estimation for exploring the potential of current and future hardware for computational and simulation tasks. Using these abstractions, we present both new and existing examples that illustrate the benefits of hybrid CV-DV processors relative to traditional DV-only hardware in computation as well as quantum simulation of physical models. Examples include algorithms for transferring states between DV and CV systems, performing the quantum Fourier transform, and simulation of lattice gauge theories. Relative to qubit-only hardware, the bosonic degrees of freedom natively available in hybrid architectures can substantially reduce the circuit complexity of simulations for physical models containing bosons. A key technique is the extension of quantum signal processing ideas to CV-DV systems. This work is intended to serve as a timely and comprehensive guide to this relatively unexplored yet promising approach to quantum computation and to provide a road map to guide future development.
Quantum computing has traditionally centered around the discrete variable paradigm. A new direction is the inclusion of continuous variable modes and the consideration of a hybrid continuous-discrete approach to quantum computing. In this paper, we discuss some of the advantages of this modality, and lay out a number of potential applications that can make use of it; these include applications from physics, chemistry, and computer science. We also briefly overview some of the algorithmic and software considerations for this new paradigm.
We introduce a framework for simulating hybrid oscillator-qubit quantum processors on qubit-only systems through position encoding. By encoding continuous-variable position and momentum wave functions into qubit amplitudes, our method efficiently simulates all Gaussian and conditional Gaussian operations -- encompassing the phase-space instruction set (beam splitter, single-qubit rotation, conditional displacement) and extending to squeezing, conditional squeezing, conditional rotation, and conditional beam splitter -- using O(log2(Γ+log(1/ε))) qubit gates per hybrid gate, where Γ is the Fock-level bound and ε is the target precision. This polylogarithmic per-gate complexity represents an exponential improvement over Fock basis encoding approaches, which require exponential quantum or classical resources in the number of qubits per mode. We provide rigorous numerical characterization of quantum Fourier transform errors for Fock-bounded states, enabling precise resource estimation for practical implementations. This work establishes that hybrid oscillator-qubit algorithms can be implemented on qubit processors with polynomial overhead, providing new insights into the computational power trade-offs between discrete-variable and hybrid continuous-discrete-variable quantum computing.
(Abridged.) Quantum computers promise to solve some problems exponentially faster than traditional computers, but we still do not fully understand why this is the case. While the most studied model of quantum computation uses qubits, which are the quantum equivalent of a classical bit, an alternative method for building quantum computers is gaining traction. Continuous-variable devices, with their infinite range of measurement outcomes, use systems such as electromagnetic fields. Given this infinite-dimensional structure, combined with the complexities of quantum physics, we are left with a natural question: when are continuous-variable quantum computers more powerful than classical devices? This thesis investigates this question by exploring the boundary of which circuits are classically simulatable and which unlock a quantum advantage over classical computers. A series of proofs are presented demonstrating the efficient simulatability of progressively more complex circuits, even those with high amounts of Wigner negativity. Specifically, circuits initiated with highly Wigner-negative Gottesman-Kitaev-Preskill states, which form a grid-like structure in phase space, can be simulated in polynomial time.