Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation" algorithm capable of harnessing this exponential advantage, that can apply polynomial transformations to the singular values of a block of a unitary, generalizing the optimal Hamiltonian simulation results of Low and Chuang [LC17a]. The proposed quantum circuits have a very simple structure, often give rise to optimal algorithms and have appealing constant factors, while typically only use a constant number of ancilla qubits.We show that singular value transformation leads to novel algorithms. We give an efficient solution to a "non-commutative" measurement problem used for efficient ground-state-preparation of certain local Hamiltonians, and propose a new method for singular value estimation. We also show how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum. Finally, as a quantum machine learning application we show how to efficiently implement principal component regression."Singular value transformation" is conceptually simple and efficient, and leads to a unified framework of quantum algorithms incorporating a variety of quantum speed-ups. We illustrate this by showing how it generalizes a number of prominent quantum algorithms, and quickly derive the following algorithms: optimal Hamiltonian simulation, implementing the Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude amplification, robust oblivious amplitude amplification, fast QMA amplification, fast quantum OR lemma, certain quantum walk results and several quantum machine learning algorithms.In order to exploit the strengths of the presented method it is useful to know its limitations too, therefore we also prove a lower bound on the efficiency of singular value transformation, which often gives optimal bounds.

Given a Hermitian operatorĤ = G|Û |G that is the projection of an oracleÛ by state |G created with oracleĜ, the problem of Hamiltonian simulation is approximating the time evolution operator e −iĤt at time t with error . We show that this can be done with query complexity O t + log (1/ ) log log (1/ ) toĜ,Û that is optimal, not just in asymptotic limits, but for all values t, . Furthermore, only 2 additional ancilla qubits are required in total, together with O(1) additional single and two-qubit gates per query. Our 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 complexity, as well as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where G|Û |G is a density matrix. A key technical result is 'qubitization' which uses controlled-Û and controlled-Ĝ to embedĤ in an invariant SU(2) subspace. A large class of operator functions ofĤ can then be computed with optimal query complexity, of which e −iĤt is a special case. d j=1 α jÛj is a linear combination of d unitaries and the |α j | sum to α -such a decomposition always exists -with an algorithm using O log (d) log (αt/ ) log log (αt/ ) ancilla qubits and only O αt log (αt/ ) log log (αt/ ) queries. Subsequently [10], an extension to d-sparse Hamiltonians was made, whereĤ has ≤ d

The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a d-sparse Hamiltonian H for time-interval t with error ϵ is O½td∥Ĥ∥ max þ log ð1=ϵÞ= log log ð1=ϵÞ, which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (i) transducing eigenvalues ofĤ into a single ancilla qubit, (ii) transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (iii) projecting this ancilla with near unity success probability. DOI: 10.1103/PhysRevLett.118.010501 Introduction.-According to Feynman, "if you want to make a simulation of nature, you'd better make it quantum mechanical, and by golly it's a wonderful problem, because it doesn't look so easy." [1]. Quantum computers are based on the physics of quantum mechanics, a fundamental tenant of nature as we know it. Thus, it seems natural to expect that the design and interpretation of quantum algorithms be heavily driven by physical intuition. The adiabatic algorithm [2,3] inspired by adiabaticity, and quantum walks [4,5] inspired by locality, are prominent examples. However, many quantum algorithms, most surprisingly those for the simulation of physical systems [6], are not as similarly insightful, and successive improvements in their complexity and analysis trend towards increasing abstraction and mathematical sophistication.Analogous to physical theories, good quantum algorithms for physics simulations should, beyond being correct, also ideally be simple. In seeking simplicity, not only is their implementation on physical machines eased, but so too could their performance and understanding be enhanced. As the essence of coherent quantum computation is the design of unitary operations with desired properties, this motivates consideration of its closest analogue in experiments: physical quantum control [7], which has a similar goal of designing quantum response functions [8].This hints at a deep connection between the design of optimal quantum algorithms and the synthesis of optimal quantum control policies. While robust time-optimal control [9,10] is certainly an established tool in quantum computing, its role is often secondary to the ends: the synthesis of computing primitives, such as Clifford

Grover's quantum search and its generalization, quantum amplitude amplification, provide a quadratic advantage over classical algorithms for a diverse set of tasks but are tricky to use without knowing beforehand what fraction λ of the initial state is comprised of the target states. In contrast, fixed-point search algorithms need only a reliable lower bound on this fraction but, as a consequence, lose the very quadratic advantage that makes Grover's algorithm so appealing. Here we provide the first version of amplitude amplification that achieves fixed-point behavior without sacrificing the quantum speedup. Our result incorporates an adjustable bound on the failure probability and, for a given number of oracle queries, guarantees that this bound is satisfied over the broadest possible range of λ. Grover's quantum search algorithm [1] provides a quadratic speedup over classical algorithms for solving a broad class of problems. Included are the many important, yet computationally prohibitive nondeterministic polynomial time (NP) problems [2], which can always be solved, albeit inefficiently, by searching the space of possible solutions. Because the problem Grover's algorithm solves is so simple to understand-given an oracle function that recognizes marked items, locate one of M such marked items among N unsorted items-its classical time complexity OðN=MÞ is obvious, making the quantum speedup that much more conclusive.Conceptually also, Grover's algorithm is compellingthe iterative application of the oracle and initial state preparation rotates from a superposition of mostly unmarked states to a superposition of mostly marked states in just Oð ffiffiffiffiffiffiffiffiffiffi ffi N=M p Þ steps [3]. This interpretation of Grover's algorithm as a rotation is very natural because the Grover iterate is a unitary operator. However, this same unitarity is also a weakness. Without knowing exactly how many marked items there are, there is no knowing when to stop the iteration. This leads to the soufflé problem [4], in which iterating too little "undercooks" the state, leaving mostly unmarked states, and iterating too much "overcooks" the state, passing by the marked states and leaving us again with mostly unmarked states.The most direct solution of the soufflé problem is to estimate M by either using full-blown quantum counting [5,6] or a trial-and-error scheme where iterates are applied an exponentially increasing number of times [5,7]. Although scaling quantumly, these strategies are unappealing for search as they work best not by monotonically amplifying marked states, but rather by getting "close enough" before resorting to classical random sampling.An alternative approach, in line with what we advocate here, is to construct, either recursively or dissipatively, operators that avoid overcooking by always amplifying marked states. Such algorithms are known as fixed-point searches. For example, running Grover's π=3 algorithm [8] or the comparable ancilla algorithm [9] longer can only ever improve its success probability....

The creation of composite quantum gates that implement quantum response functionsÛðθÞ dependent on some parameter of interest θ is often more of an art than a science. Through inspired design, a sequence of L primitive gates also depending on θ can engineer a highly nontrivialÛðθÞ that enables myriad precision metrology, spectroscopy, and control techniques. However, discovering new, useful examples ofÛðθÞ requires great intuition to perceive the possibilities, and often brute force to find optimal implementations. We present a systematic and efficient methodology for composite gate design of arbitrary length, where phase-controlled primitive gates all rotating by θ act on a single spin. We fully characterize the realizable family ofÛðθÞ, provide an efficient algorithm that decomposes a choice ofÛðθÞ into its shortest sequence of gates, and show how to efficiently choose an achievableÛðθÞ that, for fixed L, is an optimal approximation to objective functions on its quadratures. A strong connection is forged with classical discrete-time signal processing, allowing us to swiftly construct, as examples, compensated gates with optimal bandwidth that implement arbitrary single-spin rotations with subwavelength spatial selectivity.

We study the problem of simulating the time evolution of a lattice Hamiltonian, where the qubits are laid out on a lattice and the Hamiltonian only includes geometrically local interactions (i.e., a qubit may only interact with qubits in its vicinity). This class of Hamiltonians is very general and encompasses essentially all physical Hamiltonians.Our algorithm simulates the time evolution of such a Hamiltonian on n qubits for time T up to error using O(nT polylog(nT / )) gates with depth O(T polylog(nT / )). Our algorithm is the first simulation algorithm that achieves gate cost quasilinear in nT and polylogarithmic in 1/ . Our algorithm also readily generalizes to time-dependent Hamiltonians and yields an algorithm with similar gate count for any piecewise slowly varying time-dependent bounded local Hamiltonian.We also prove a matching lower bound on the gate count of such a simulation, showing that any quantum algorithm that can simulate a piecewise constant bounded local Hamiltonian in one dimension to constant error requires Ω(nT ) gates in the worst case. The lower bound holds even if we only require the output state to be correct on local measurements. To our best knowledge, this is the first nontrivial lower bound on the gate complexity of the simulation problem.Our algorithm is based on a decomposition of the time-evolution unitary into a product of small unitaries using Lieb-Robinson bounds. In the appendix, we prove a Lieb-Robinson bound tailored to Hamiltonians with small commutators between local terms, giving zero Lieb-Robinson velocity in the limit of commuting Hamiltonians. This improves the performance of our algorithm when the Hamiltonian is close to commuting.

In this paper we outline the extension of recently introduced the sub-system embedding subalgebras coupled cluster (SES-CC) formalism to the unitary CC formalism. In analogy to the standard single-reference SES-CC formalism, its unitary CC extension allows one to include the dynamical (outside the active space) correlation effects in an SES induced complete active space (CAS) effective Hamiltonian. In contrast to the standard single-reference SES-CC theory, the unitary CC approach results in a Hermitian form of the effective Hamiltonian. Additionally, for the double unitary CC formalism (DUCC) the corresponding CAS eigenvalue problem provides a rigorous separation of external cluster amplitudes that describe dynamical correlation effects -used to define the effective Hamiltonian -from those corresponding to the internal (inside the active space) excitations that define the components of eigenvectors associated with the energy of the entire system. The proposed formalism can be viewed as an efficient way of downfolding many-electron Hamiltonian to the low-energy model represented by a particular choice of CAS. In principle, this technique can be extended to any type of complete active space representing an arbitrary energy window of a quantum system. The Hermitian character of low-dimensional effective Hamiltonians makes them an ideal target for several types of full configuration interaction (FCI) type eigensolvers. As an example, we also discuss the algebraic form of the perturbative expansions of the effective DUCC Hamiltonians corresponding to composite unitary CC theories and discuss possible algorithms for hybrid classical and quantum computing.

An important step in building a quantum computer is calibrating experimentally implemented quantum gates to produce operations that are close to ideal unitaries. The calibration step involves estimating the systematic errors in gates and then using controls to correct the implementation. Quantum process tomography is a standard technique for estimating these errors but is both time consuming (when one wants to learn only a few key parameters) and usually inaccurate without resources such as perfect state preparation and measurement, which might not be available. With the goal of efficiently and accurately estimating specific errors using minimal resources, we develop a parameter estimation technique, which can gauge key systematic parameters (specifically, amplitude and off-resonance errors) in a universal single-qubit gate set with provable robustness and efficiency. In particular, our estimates achieve the optimal efficiency, Heisenberg scaling, and do so without entanglement and entirely within a single-qubit Hilbert space. Our main theorem making this possible is a robust version of the phase estimation procedure of Higgins et al.