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 discuss the problem of counting the maximum number of distinct states that an isolated physical system can pass through in a given period of time---its maximum speed of dynamical evolution. Previous analyses have given bounds in terms of the standard deviation of the energy of the system; here we give a strict bound that depends only on E-E0, the system's average energy minus its ground state energy. We also discuss bounds on information processing rates implied by our bound on the speed of dynamical evolution. For example, adding one Joule of energy to a given computer can never increase its processing rate by more than about 3x10^33 operations per second.Comment: 14 pages, no figures, LaTex2e (elsart). This is the published version, which includes brief semi-classical and relativistic discussions not included in the original preprin

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Quantum speed limits set an upper bound to the rate at which a quantum system can evolve. Adopting a phase-space approach we explore quantum speed limits across the quantum to classical transition and identify equivalent bounds in the classical world. As a result, and contrary to common belief, we show that speed limits exist for both quantum and classical systems. As in the quantum domain, classical speed limits are set by a given norm of the generator of time evolution.The multi-faceted nature of time makes its treatment challenging in the quantum world [1,2]. Nonetheless, the understanding of time-energy uncertainty relations is somewhat privileged [3,4]. To a great extent, this is due to their reformulation in terms of quantum speed limits (QSL) concerning the ability to distinguish two quantum states connected via time evolution. While QSL provide fundamental constraints to the pace at which quantum systems can change, a plethora of applications have been found that well extend beyond the realm of quantum dynamics. Indeed, QSL provide limits to the computational capability of physical devices [5], the performance of quantum thermal machines in finite-time thermodynamics [6,7], parameter estimation in quantum metrology [8,9], quantum control [10][11][12][13][14], the decay of unstable quantum systems [15][16][17][18] and information scrambling [19], among other examples [3,4,20].Specifically, QSL are derived as upper bounds to the rate of change of the fidelity F(τ) = | ψ 0 |ψ τ | 2 ∈ [0, 1] between an initial quantum state |ψ 0 and the corresponding time-evolving state |ψ τ =Û(τ, 0)|ψ 0 , whereÛ(τ, 0) is the time-evolution operator. More generally quantum states need not be pure, and given two density matrices ρ 0 and ρ τ =Û(τ, 0)ρ 0Û (τ, 0) † the fidelity readsThe fidelity is useful to define a metric between quantum states in Hilbert space, known as the Bures angle, [24,25] This gives a geometric interpretation of speed limit as the minimum time required to sweep out the angle L (ρ 0 , ρ τ ) under a given dynamics [26]. For unitary processes, two seminal results are known. The Mandelstam-Tamm bound estimates the speed of evolution in terms of the energy dispersion of the initial state [15, 16, 21-23, 25, 27]. Its original derivation relies on the Heisenberg uncertainty relation. The second seminal result is named after Margolus and Levitin, and provides an upper bound to the speed of evolution in term of the difference between the mean energy and the ground state energy [28,29]. Its original derivation relies on the study of the survival amplitude ψ 0 |ψ τ . These bounds can be extended to driven and open quantum systems [30][31][32][33][34][35]. In addition, the two bounds can be unified [29] so that the time of evolution τ required to sweep an angle L (ρ 0 , ρ τ ) is lower bounded bywhere E 0 is the ground state of the system, E is its mean energy, and ∆E denotes the energy dispersion. Note however that there is an infinite family of bounds in terms of higher order moments of the energy of the system [3...