We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (that maps Boolean values $(x,y)$ to $(x,x \oplus y)$) is universal in the sense that all unitary operations on arbitrarily many bits $n$ (U($2^n$)) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two-and three-bit quantum gates, the asymptotic number required for $n$-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary $n$-bit unitary operations.Comment: 31 pages, plain latex, no separate figures, submitted to Phys. Rev. A. Related information on http://vesta.physics.ucla.edu:7777

We investigate the concept of quantum secret sharing. In a ((k, n)) threshold scheme, a secret quantum state is divided into n shares such that any k of those shares can be used to reconstruct the secret, but any set of k − 1 or fewer shares contains absolutely no information about the secret. We show that the only constraint on the existence of threshold schemes comes from the quantum "no-cloning theorem", which requires that n < 2k, and, in all such cases, we give an efficient construction of a ((k, n)) threshold scheme. We also explore similarities and differences between quantum secret sharing schemes and quantum error-correcting codes. One remarkable difference is that, while most existing quantum codes encode pure states as pure states, quantum secret sharing schemes must use mixed states in some cases. For example, if k ≤ n < 2k − 1 then any ((k, n)) threshold scheme must distribute information that is globally in a mixed state.Suppose that the president of a bank wants to give access to a vault to three vice presidents who are not entirely trusted. Instead of giving the combination to any one individual, it may be desirable to distribute information in such a way that no vice president alone has any knowledge of the combination, but any two of them can jointly determine the combination. In 1979, Blakely [1] and Shamir [2] addressed a generalization of this problem, by showing how to construct schemes that divide a secret into n shares such that any k of those shares can be used to reconstruct the secret, but any set of k − 1 or fewer shares contains absolutely no information about the secret. This is called a (k, n) threshold scheme, and is a useful tool for designing cryptographic key management systems. Now, consider a generalization of such schemes to the setting of quantum information, where the secret is an arbitrary unknown quantum state. Salvail [3] (see also [4]) obtained a method to divide an unknown qubit into two shares, each of which individually contains no information about the qubit, but which jointly can be used to reconstruct the qubit. Hillery, Bužek, and Berthiaume [4] proposed a method for implementing some clas- * Email: cleve@cpsc.ucalgary.ca † Email: gottesma@t6-serv.lanl.gov ‡ Email: hkl@hplb.hpl.hp.com sical threshold schemes that uses quantum information to transmit the shares securely in the presence of eavesdroppers. Define a ((k, n)) threshold scheme, with k ≤ n, as a method to encode and divide an arbitrary secret quantum state (which is given but not, in general, explicitly known) into n shares with the following two properties. First, from any k or more shares the secret quantum state can be perfectly reconstructed. Second, from any k − 1 or fewer shares, no information at all can be deduced about the secret quantum state. Formally, this means that the reduced density matrix of these k − 1 shares (with the other shares traced out) is independent of the value of the secret. Each share can consist of any number of qubits (or higher-dimensional states), and not all shares nee...

Quantum computers use the quantum interference of different computational paths to enhance correct outcomes and suppress erroneous outcomes of computations. A common pattern underpinning quantum algorithms can be identified when quantum computation is viewed as multi-particle interference. We use this approach to review (and improve) some of the existing quantum algorithms and to show how they are related to different instances of quantum phase estimation. We provide an explicit algorithm for generating any prescribed interference pattern with an arbitrary precision.

Classical fingerprinting associates with each string a shorter string (its fingerprint), such that, with high probability, any two distinct strings can be distinguished by comparing their fingerprints alone. The fingerprints can be exponentially smaller than the original strings if the parties preparing the fingerprints share a random key, but not if they only have access to uncorrelated random sources. In this paper we show that fingerprints consisting of quantum information can be made exponentially smaller than the original strings without any correlations or entanglement between the parties: we give a scheme where the quantum fingerprints are exponentially shorter than the original strings and we give a test that distinguishes any two unknown quantum fingerprints with high probability. Our scheme implies an exponential quantum/classical gap for the equality problem in the simultaneous message passing model of communication complexity. We optimize several aspects of our scheme.

We present an efficient quantum algorithm for simulating the evolution of a sparse Hamiltonian H for a given time t in terms of a procedure for computing the matrix entries of H. In particular, when H acts on n qubits, has at most a constant number of nonzero entries in each row/column, and H is bounded by a constant, we may select any positive integer k such that the simulation requires O((log * n)t 1+1/2k ) accesses to matrix entries of H. We show that the temporal scaling cannot be significantly improved beyond this, because sublinear time scaling is not possible.

We develop the concept of a unitary t-design as a means of expressing operationally useful subsets of the stochastic properties of the uniform (Haar) measure on the unitary group U (2 n ) on n qubits. In particular, sets of unitaries forming 2-designs have wide applicability to quantum information protocols. We devise an O(n)-size in-place circuit construction for an approximate unitary 2-design. We then show that this can be used to construct an efficient protocol for experimentally characterizing the fidelity of a quantum process on n qubits with quantum circuits of size O(n) without requiring any ancilla qubits, thereby improving upon previous approaches.

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...

We examine the number