We obtain sufficient conditions for the efficient simulation of a continuous variable quantum algorithm or process on a classical computer. The resulting theorem is an extension of the Gottesman-Knill theorem to continuous variable quantum information. For a collection of harmonic oscillators, any quantum process that begins with unentangled Gaussian states, performs only transformations generated by Hamiltonians that are quadratic in the canonical operators, and involves only measurements of canonical operators (including finite losses) and suitable operations conditioned on these measurements can be simulated efficiently on a classical computer.

We show that higher-dimensional versions of qubits, or qudits, can be encoded into spin systems and into harmonic oscillators, yielding important advantages for quantum computation. Whereas qubit-based quantum computation is adequate for analyses of quantum vs classical computation, in practice qubits are often realized in higher-dimensional systems by truncating all but two levels, thereby reducing the size of the precious Hilbert space. We develop natural qudit gates for universal quantum computation, and exploit the entire accessible Hilbert space. Mathematically, we give representations of the generalized Pauli group for qudits in coupled spin systems and harmonic oscillators, and include analyses of the qubit and the infinite-dimensional limits.PACS numbers: 03.67. Lx, Quantum computation may be able to perform certain tasks more efficiently than a classical computer; for example, Shor's algorithm [1] for factoring prime numbers on a quantum computer is exponentially faster than any known algorithm on a classical computer. The standard model of a quantum computer involves coupling together two-level quantum systems (qubits) such that the Hilbert space of the system grows exponentially in the number of qubits.A major obstacle to universal quantum computing is the limit on the number of coupled qubits that can be achieved in a physical system [2]. The use of ddimensional, or qudit, quantum computing enables a much more compact and efficient information encoding than for qubit computing. Qudit quantum information processing employs fewer coupled quantum systems: a considerable advantage for the experimental realization of quantum computing. The harmonic oscillator is a system that naturally provides qudits as quanta in its energy spectrum. Qubits are obtained by restricting the dynamics to just two of these quanta, namely the vacuum state |0 and the first excited state |1 ; e.g., photons in cavity QED [3] and interferometry [4]. However, the control of entanglement in larger Hilbert spaces is now feasible (e.g., orbital angular momentum states of photons [5]). Our aim is to show that the restriction to two-dimensional Hilbert spaces is not necessary and that higher-dimensional Hilbert spaces are an advantage, particularly when the number of achievable coupled systems is limited and entanglement between systems with larger Hilbert spaces is physically possible.A quantum computer also requires gates, realized as the unitary evolution under some Hamiltonian. For qubits, a universal set of gates is given by arbitrary SU(2) rotations of a single qubit along with some nonlinear coupling transformation between adjacent qubits generated by a two-qubit Hamiltonian [6]. For qudit quantum computation, the issue of creating a universal set of gates is more involved. In particular, it is not possible to treat coupled qudits as a collection of qubits, because (typically) one does not have access to "pairwise" Hamiltonians between two arbitrary levels of coupled qudits. For example, in a system of coupled oscillators realize...

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Remarkable experimental advances in quantum computing are exemplified by recent announcements of impressive average gate fidelities exceeding 99.9% for single-qubit gates and 99% for two-qubit gates. Although these high numbers engender optimism that fault-tolerant quantum computing is within reach, the connection of average gate fidelity with fault-tolerance requirements is not direct. Here we use reported average gate fidelity to determine an upper bound on the quantum-gate error rate, which is the appropriate metric for assessing progress towards fault-tolerant quantum computation, and we demonstrate that this bound is asymptotically tight for general noise. Although this bound is unlikely to be saturated by experimental noise, we demonstrate using explicit examples that the bound indicates a realistic deviation between the true error rate and the reported average fidelity. We introduce the Pauli distance as a measure of this deviation, and we show that knowledge of the Pauli distance enables tighter estimates of the error rate of quantum gates. lb 0

Topological quantum states are characterized by nonlocal invariants, and their detection is intrinsically challenging. Various strategies have been developed to study topological Hamiltonians through their equilibrium states. We present a fundamentally new, high-precision dynamical approach, revealing topology through the unitary evolution after a quench from a topological trivial initial state with a two-dimensional Chern band realized in an ultracold 87 Rb atom gas. The emerging ring structure in the spin dynamics uniquely determines the Chern number for the post-quench band and enables probing the full phase diagram of the band topology with high precision. Besides, we also measure the topological band gap and the domain wall structure dynamically formed in the momentum space during the long-term evolution. Our dynamical approach provides a way towards observing a universal bulk-ring correspondence, and has broad applications in exploring topological quantum matter.The discovery of the quantum Hall effect introduced a new fundamental concept, topological quantum phase, to condensed-matter physics [1,2]. In contrast to symmetry-breaking quantum phases that are characterized by local order parameters in the Landau paradigm, topological quantum matter is classified by nonlocal topological invariants [3], which usually do not directly correspond to the local physical observables. In consequence the detection of topological states is intrinsically challenging. In solid-state experiments, various strategies have been developed and great success has been achieved in the discovery of topological quantum matter like topological insulators [4][5][6][7] and semimetals [8,9]. For instance, transport measurements and angle-resolved photoemission spectroscopy are used to detect gapless boundary modes that reflect the bulk topology [10,11]. In some circumstances, these strategies do not provide fully unambiguous evidences for topological quantum phases, as they do not directly measure topological numbers. An important example is topological superconductivity, which supports a kind of exotic non-Abelian quasiparticle called Majorana modes [12][13][14][15] and remains to be rigorously confirmed by experiment.To precisely detect the topology for an ultracold-atom system can be more challenging, whereas the extensive tool box of manipulating and probing ultracold atoms may offer distinct new strategies for measurement. For a one-dimensional (1D) Su-Schrieffer-Heeger model simulated with a 1D double well lattice, the band topology can be determined by measuring the Zak phase [16]. Furthermore, the bulk topology of a 2D Chern insulator, characterized by Chern invariants, can be observed by Hall transport studies [17,18], by Berry curvature mapping [19], and by a minimal measurement strategy [20] of imaging the spin texture at symmetric Bloch momenta [21]. The Chern invariants are precisely detectable only in a few special cases, e.g. when the bulk band is flat [17] or the system is of inversion symmetry [20,21]. Nevertheless, the ...

Finding exponential separation between quantum and classical information tasks is like striking gold in quantum information research. Such an advantage is believed to hold for quantum computing but is proven for quantum communication complexity. Recently, a novel quantum resource called the quantum switch-which creates a coherent superposition of the causal order of events, known as quantum causality-has been harnessed theoretically in a new protocol providing provable exponential separation. We experimentally demonstrate such an advantage by realizing a superposition of communication directions for a two-party distributed computation. Our photonic demonstration employs d-dimensional quantum systems, qudits, up to d = 2 13 dimensions and demonstrates a communication complexity advantage, requiring less than (0.696 ± 0.006) times the communication of any causally ordered protocol. These results elucidate the crucial role of the coherence of communication direction in achieving the exponential separation for the one-way processing task, and open a new path for experimentally exploring the fundamentals and applications of advanced features of indefinite causal structures.Computation by separated parties with minimal communication is the focus of communication complexity, which has applications to distributed computing, very-large-scale integration, streaming algorithms, and more [1]. For quantum information, communication complexity is especially exciting as exponential quantum-classical gaps can be proven [2][3][4][5][6][7][8][9]. By contrast, exponential quantum-classical gaps for computation tasks such as factorization [10] depend on the best-known classical algorithm, and thus are strongly believed but not rigorously proven. Experimentally, quantum communication complexity has been studied in proof of principle for the quantum fingerprinting protocol [11][12][13] and beyond [1,14,15].The quantum switch provides a new communication complexity tool that leads to another instance of exponential quantum advantage [16]. The quantum switch is a device where a control qubit determines the order in which two transformations are performed on a target system [17,18]. When the control is in a superposition of logical states, the order of the operations is causally indefinite; i.e., there is a superposition of the ordering of target operations. The quantum switch has broad relevance in the context of quantum causality [19] including applications to studies of quantum gravity [19][20][21][22], communication complexity [16, * These authors contributed equally to this work.2 23], witnessing causality [17,[24][25][26][27][28] and deciding whether a given indefinite causal order is physical [29,30]. In quantum computing, the quantum switch can reduce the query complexity for some tasks compared to causally ordered protocols [18,31] -this advantage has been demonstrated for single-qubit control and single-qubit target circuits [32]. In quantum communication, the quantum switch enhances the communication rate beyond the limits of ...

We introduce the entangled coherent-state representation, which provides a powerful technique for efficiently and elegantly describing and analyzing quantum optics sources and detectors while respecting the photon-number superselection rule that is satisfied by all known quantum optics experiments. We apply the entangled coherent-state representation to elucidate and resolve the long-standing puzzles of the coherence of a laser output field, interference between two number states, and dichotomous interpretations of quantum teleportation of coherent states.

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