We describe a simple, efficient method for simulating Hamiltonian dynamics on a quantum computer by approximating the truncated Taylor series of the evolution operator. Our method can simulate the time evolution of a wide variety of physical systems. As in another recent algorithm, the cost of our method depends only logarithmically on the inverse of the desired precision, which is optimal. However, we simplify the algorithm and its analysis by using a method for implementing linear combinations of unitary operations together with a robust form of oblivious amplitude amplification. DOI: 10.1103/PhysRevLett.114.090502 PACS numbers: 03.67.Ac, 89.70.Eg One of the main motivations for quantum computers is their ability to efficiently simulate the dynamics of quantum systems [1], a problem that is apparently hard for classical computers. Since the mid-1990s, many algorithms have been developed to simulate Hamiltonian dynamics on a quantum computer [2][3][4][5][6][7][8][9][10][11][12], with applications to problems such as simulating spin models [13] and quantum chemistry [14][15][16][17]. While it is now well known that quantum computers can efficiently simulate Hamiltonian dynamics, ongoing work has improved the performance and expanded the scope of such simulations.Recently, we introduced a new approach to Hamiltonian simulation with exponentially improved performance as a function of the desired precision [18]. Specifically, we presented a method to simulate a d-sparse, n-qubit Hamiltonian H acting for time t > 0, within precision ϵ > 0, using O(τ logðτ=ϵÞ= log logðτ=ϵÞ) queries to H and O(nτlog 2 ðτ=ϵÞ= log logðτ=ϵÞ) additional two-qubit gates, where τ ≔ d 2 ∥H∥ max t. This dependence on ϵ is exponentially better than all previous approaches to Hamiltonian simulation, and the number of queries to H is optimal [18]. (For simplicity, we refer to combinations of logarithms like those in the above expressions as logarithmic.) Roughly speaking, doubling the number of digits of accuracy only doubles the complexity.The simulation algorithm of [18] is indirect, appealing to an unconventional model of query complexity. In this Letter, we describe and analyze a simplified approach to Hamiltonian simulation with the same cost as the method of [18]. The new approach is easier to understand, and the reason for the logarithmic cost dependence on ϵ is immediate. The new approach decomposes the Hamiltonian into a linear combination of unitary operations. Unlike the algorithm of [18], these terms need not be self-inverse, so the algorithm is efficient for a larger class of Hamiltonians. The new approach is also simpler to analyze: we give a selfcontained presentation in four pages.The main idea of the new approach is to implement the truncated Taylor series of the evolution operator. Similar to previous approaches for implementing linear combinations of unitary operators [12,13], the various terms of the Taylor series can be implemented by introducing an ancillary superposition and performing controlled operations. The time e...

Harrow, Hassidim, and Lloyd showed that for a suitably specified N × N matrix A and Ndimensional vector b, there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of equations A x = b. If A is sparse and well-conditioned, their algorithm runs in time poly(log N, 1/ǫ), where ǫ is the desired precision in the output state. We improve this to an algorithm whose running time is polynomial in log(1/ǫ), exponentially improving the dependence on precision while keeping essentially the same dependence on other parameters. Our algorithm is based on a general technique for implementing any operator with a suitable Fourier or Chebyshev series representation. This allows us to bypass the quantum phase estimation algorithm, whose dependence on ǫ is prohibitive.

Quantum state tomography-deducing quantum states from measured data-is the gold standard for verification and benchmarking of quantum devices. It has been realized in systems with few components, but for larger systems it becomes unfeasible because the number of measurements and the amount of computation required to process them grows exponentially in the system size. Here, we present two tomography schemes that scale much more favourably than direct tomography with system size. one of them requires unitary operations on a constant number of subsystems, whereas the other requires only local measurements together with more elaborate post-processing. Both rely only on a linear number of experimental operations and post-processing that is polynomial in the system size. These schemes can be applied to a wide range of quantum states, in particular those that are well approximated by matrix product states. The accuracy of the reconstructed states can be rigorously certified without any a priori assumptions.

Physical systems, characterized by an ensemble of interacting constituents, can be represented and studied by different algebras of operators ͑observables͒. For example, a fully polarized electronic system can be studied by means of the algebra generated by the usual fermionic creation and annihilation operators or by the algebra of Pauli ͑spin-1/2͒ operators. The Jordan-Wigner isomorphism gives the correspondence between the two algebras. As we previously noted, similar isomorphisms enable one to represent any physical system in a quantum computer. In this paper we evolve and exploit this fundamental observation to simulate generic physical phenomena by quantum networks. We give quantum circuits useful for the efficient evaluation of the physical properties ͑e.g., the spectrum of observables or relevant correlation functions͒ of an arbitrary system with Hamiltonian H.

We introduce a generalization of entanglement based on the idea that entanglement is relative to a distinguished subspace of observables rather than a distinguished subsystem decomposition. A pure quantum state is entangled relative to such a subspace if its expectations are a proper mixture of those of other states. Many information-theoretic aspects of entanglement can be extended to the general setting, suggesting new ways of measuring and classifying entanglement in multipartite systems. By going beyond the distinguishable-subsystem framework, generalized entanglement also provides novel tools for probing quantum correlations in interacting many-body systems.PACS numbers: 03.67.Mn, 03.65.Ud, Entanglement is a uniquely quantum phenomenon whereby a pure state of a composite quantum system may cease to be determined by the states of its constituent subsystems [1]. Entangled pure states are those that have mixed subsystem states. To determine an entangled state requires knowledge of the correlations between the subsystems. As no pure state of a classical system can be correlated, such correlations are intrinsically non-classical, as strikingly manifested by the violation of local realism and Bell's inequalities [2]. In the science of quantum information processing (QIP), entanglement is regarded as the defining resource for quantum communication and an essential feature needed for unlocking the power of quantum computation. However, in spite of intensive investigation, a complete understanding of entanglement is far from being reached.To unambiguously define entanglement requires a preferred partition of the overall system into subsystems. In conventional QIP scenarios, subsystems are associated with spatially separated "local" parties, which legitimates the distinguishability assumption implicit in standard entanglement theory. However, because quantum correlations are at the heart of many physical phenomena, it would be desirable for a notion of entanglement to be useful in contexts other than QIP. Strongly interacting quantum systems offer compelling examples of situations where the usual subsystem-based view is inadequate. Whenever indistinguishable particles are sufficiently close to each other, quantum statistics forces the accessible state space to be a proper subspace of the full tensor product space, and exchange correlations arise that are not a usable resource in the usual QIP sense. Thus, the natural identification of particles with preferred subsystems becomes problematic. Even if a distinguishable-subsystem structure may be associated to degrees of freedom different from the original particles (such as a set of modes [3]), inequivalent factorizations may occur on the same footing. Finally, the introduction of quasiparticles, or the purposeful transformation of the algebraic language used to analyze the system [4], may further complicate the choice of preferred subsystems.While efforts are under way to obtain entanglement-like notions for bosons and fermions [3,5] and to study entanglement in quant...

We consider the manifold of all quantum many-body states that can be generated by arbitrary time-dependent local Hamiltonians in a time that scales polynomially in the system size, and show that it occupies an exponentially small volume in Hilbert space. This implies that the overwhelming majority of states in Hilbert space are not physical as they can only be produced after an exponentially long time. We establish this fact by making use of a time-dependent generalization of the Suzuki-Trotter expansion, followed by a counting argument. This also demonstrates that a computational model based on arbitrarily rapidly changing Hamiltonians is no more powerful than the standard quantum circuit model.The Hilbert space of a quantum system is big-its dimension grows exponentially with the number of particles it contains. Thus, parametrizing a generic quantum state of N particles requires an exponential number of real parameters. Fortunately, the states of many physical systems of interest appear to occupy a tiny sub-manifold of this gigantic space. Indeed, the essential physical features of many systems can be explained by variational states specified with a small number of parameters. Well known examples include the BCS state for superconductivity [1], Laughlin's state for fractional quantum Hall liquids [2], tensor network states occurring in real-space renormalization methods [3]. In these cases, the number of parameters scales only polynomially with N .In this Letter, we attempt to define the class of physical states of a many-body quantum system with local Hilbert spaces of bounded dimensions, and prove that they represent an exponentially small sub-manifold of the Hilbert space. We say that a state is physical if it can be reached, starting in some fiducial state (e.g. a ferromagnetic state, or the vacuum), by an evolution generated by any time-dependent quantum many-body Hammiltonian, with the constraint that 1) the Hamiltonian is local in the sense that it is the sum of terms each acting on at most k bodies for some constant k independent of N , and 2) the duration of the evolution scales at most as a polynomial in the number of particles in the system. The assumption about the initial fiducial state is artificial; we could alternatively define the class of physical evolutions for quantum many-body systems as the ones generated by Hamiltonians obeying constraints 1 and 2, and would reach the same conclusions.The second constraint is very much reminiscent of the way complexity classes are defined in theoretical computer science, where that central object of study is the scaling of the time required to solve a problem as a function of its input size. The classical analogue for the problem that we address is a well known counting argument of Shannon [4] demonstrating that the number of boolean functions of N bits scales doubly exponentially (as 2 2 N ), with the consequence that no efficient (i.e. polynomial) algorithm can exist to compute the overwhelming majority of those functions. Indeed, the number of differe...

We introduce a generalized Gaudin Lie algebra and a complete set of mutually commuting quantum invariants allowing the derivation of several families of exactly solvable Hamiltonians. Different Hamiltonians correspond to different representations of the generators of the algebra. The derived exactly-solvable generalized Gaudin models include the Hamiltonians of Bardeen-Cooper-Schrieffer, Suhl-Matthias-Walker, Lipkin-Meshkov-Glick, the generalized Dicke and atom-molecule, the nuclear interacting boson model, a new exactly-solvable Kondo-like impurity model, and many more that have not been exploited in the physics literature yet.

We provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Specifically, we show that a d-sparse Hamiltonian H acting on n qubits can be simulated for time t with precision ǫ using O τ log(τ /ǫ) log log(τ /ǫ) queries and O τlog log(τ /ǫ) n additional 2-qubit gates, where τ = d 2 H max t. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. Our algorithm is based on a significantly improved simulation of the continuous-and fractional-query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. We also simplify the analysis of this conversion, avoiding the need for a complex fault correction procedure. Our simplification relies on a new form of "oblivious amplitude amplification" that can be applied even though the reflection about the input state is unavailable. Finally, we prove new lower bounds showing that our algorithms are optimal as a function of the error.