Lattice Gas Superfluid

Transcription

1 This material has been cleared for public release by ESC/PA, 8 Nov 96, ESC HO Lattice Gas Superfluid Jeffrey Yepez US Air Force Phillips Laboratory, Hanscom Air Force Base, Massachusetts 073, USA Dated: August 29, 996 A quantum computer, with controlled decoherence, constructed out of a lattice based array of qubits undergoing a particular local and unitary evolution is effectively a Bose condensate. Presented is a coupled lattice gas system, a quantum lattice gas and a classical lattice gas in mutual contact through external potentials. The coupled lattice gas system behaves like liquid 4 He where it is a quantum fluid at a finite temperature below the λ-point. I. INTRODUCTION Ever since Feynman s conjecture in 982 that it might be possible to do exactly the same as nature by simulating quantum mechanical systems on a quantum computer, many imagined how a quantum computer might actually work. A good starting point is reversible computing. 2 With electronic computer design philosophy, research has focused on quantum gate counterparts of well known universal reversible logic gates, for instance the two-input/two-output quantum XOR gate. 3 The plan is to use the simplest universal quantum gates in networks to fashion arbitrary n-bit unitary operators. Quantum computation aims to exploit the superposition of states as a practical means of parallel computing. But this levies a high demand for coherence of the quantum computer s wavefunction, avoiding its entanglement with the external world. Consequently, this motivates developing robust and scalable error correction techniques, 4 believed crucial for the enterprise to continue. 5 7 Because of stringent demands for quantum coherence, prospects for any foreseeable quantum computer, if at all constructible, consider only a very small number of qubits. Shor s scheme for factoring numbers 8 has stirred much interest. It has motivated experimentalists to implement individual quantum gates, for example using nuclear magnetic resonance spectroscopy where a few nuclear spins in each molecule of a liquid sample embody quantum bits, 9 qubits for short. Yet the prospect of a quantum computer encompassing a vast number of qubits, packed perhaps at solidstate densities, is considered here, but one with a built-in mechanism for decohering the quantum computer s wavefunction in a controlled and periodic fashion. Imagine qubits in a lattice based array undergoing local unitary All factors are superposed in a quantum computer s register in polynomial time. A modulus operation is applied to all factors simultaneously producing a periodic function. The correct factor corresponds to the period of this function. A discrete Fourier transform is taken. Consequently, the peak in the power spectrum of the transformed data locates the factor. An amount of memory exponential in the number of bits of the composite number is not needed to perform the discrete transform. In this way a massively parallel search is accomplished in polynomial time. evolution with collide and stream partitions. 0 Such a system is known as a quantum lattice gas. 3 Imagine further that coarse-grained measurements of the quantum computer s wavefunction are periodically made causing the wavefunction to periodically collapse. Such a system might be termed a quantum lattice gas with controlled decoherence. Physical systems can behave manifestly quantum mechanically at macroscopic scales at finite temperatures the most well known example being the Helium II phase of 4 He-isotopes that obey the one-body Schrödinger equation below the λ-point. So too can a quantum lattice gas, with a suitably chosen potential function, manifest one-body quantum mechanical behavior in its scaling limit. Helium II is a quantum fluid and as such is ripe for modelling with a quantum computer. Using only the oneparticle sector of the quantum Hilbert space is necessary. The emergent macroscopic equations of motion are those of the superfluid equations of Landau s phenomenological two fluid model, an entirely classical system. In particular, the emergent hydrodynamic equation of motion of the quantum lattice gas is the Euler equation, describing the inviscid superfluid component of the flow. Note that it is not at all suggested that the quantum lattice gas presented here is useful for high Reynolds number flows. 2 The essential characteristics of the quantum lattice gas model presented in this paper are: quantum superposition between qubit states at remote sites is not necessary because of the locality of the quantum lattice gas algorithm; and 2 long coherence times between locally interacting qubit are not necessary either. A review is given of Helium II 4,5 with emphasis on the hydrodynamic twofluid model due to Landau. 6 Next a theory for a quantum lattice gas is given based on a collision matrix in direct analogy with the classical lattice gas theory; a local unitary evolution matrix is chosen so the emergent behavior of the quantum lattice gas is like a Schrödinger wave in nonrelativistic quantum mechanics. The external potential in the Schrödinger wave equation is taken to be the chemical potential of Helium II. In this way the quantum lattice gas in the scaling limit behaves as a Bose condensate manifesting emergent 2 The subject of simulating Navier-Stokes flow in a quantum computer is treated in a subsequent paper.

2 2 superfluidity. A classical lattice gas with nonlocal interactions is tailored to model the viscous normalfluid part of finite temperature Helium II. The interparticle potential of the classical lattice gas depends on the density of the quantum system measured periodically in time this induces a short coherence time. So through the action of the chemical potential and the interparticle potential the quantum lattice gas and the classical lattice gas are coupled. An additional species in the classical lattice gas is used as a passive scalar to model the entropy field of Helium II. The coupled lattice gas system is analyzed and exhibits first sound density waves at constant temperature and second sound entropy waves at constant pressure as occurs in Helium II. II. HELIUM II The isotope of Helium, 4 He, with atomic mass of four, is known to behave as a quantum liquid at low temperatures. The 4 He atom has integer spin and is a bosonic particle. The boiling point of 4 He occurs at 4.2 K, and below this temperature 4 He is a normal viscous liquid. Yet at still lower temperatures, unlike other liquids, 4 He does not freeze into a spatially organized solid phase. The interatomic force between 4 He particles is weak enough to allow 4 He to remain in the liquid phase at low enough temperatures where quantum effects can dominate and give rise to an exotic state of matter. An order-disorder phase separation does occur at 2.9 K, the so-called λ- point, but remarkably, it is a second-order phase transition in momentum space. This ordered phase in momentum space is called Helium II, the superfluid phase. The hydrodynamic properties of Helium II are fascinating, in particular the property of flow through narrow capillaries without any frictional resistance and quantum barrier tunneling manifested at macroscopic scales. Helium II behaves as a coupled two fluid system, where one fluid is a normal viscous fluid and the other is a superfluid. The underlying microscopic picture for its two fluid behavior is the following. Since Helium II is comprised of 4 He bosons, at zero temperature these particles can all occupy the same ground state energy level. This state of matter is called a Bose condensate. At finite temperatures below the λ-point, thermal excitations are created, depleting the Bose condensate, yet a macroscopic number of the 4 He particles remain in the condensate. In Helium II there are two types of thermal excitations: long wavelength phonons and shorter wavelength rotons. These excitations are quasiparticles, and away from the λ-point, are weakly interacting. They have an effective mass and transport momentum diffusively. As a collective system, the thermal excitations behave as a viscous fluid accounting for any convection and kinematic shear in Helium II. So the normal viscous part of Helium II is a gas of quasiparticles, while the superfluid part of Helium II is a Bose condensate that only transports momentum coherently. In the incompressible fluid regime of Helium II, the coherent motion of the Bose condensate is described by a wavefunction of the following form ψx = ψ e isx/. Its evolution is governed by a Schrödinger equation where the external potential is the chemical potential of the Helium II quantum fluid i t ψ = 2 2m 4 2 ψ + µψ. 2 In this manner, through the chemical potential, the dynamics of the normal and superfluid parts of the Helium II are coupled. The explicit form of µ for Helium II is given below. Following a usual quantum mechanical development, the probability current is a function of the wave function itself j prob i ψ = i ψ i ψ ψ i ψ. 3 2m 4 This implies the Helium II supercurrent density is related to the gradient of the phase of the wavefunction j s i m 4 j prob i = ψ 2 i S + j prob i, 4 where j prob i j prob i ψ is the compressible part of the superfluid current density. The condition for conservation of probability t ψ ψ + i j prob i = 0 5 then becomes the continuity equation of hydrodynamics provided the mass density of the superfluid is identified as the square of the amplitude ρ s m 4 ψ. 2 It follows that the superfluid s hydrodynamics flow velocity is v s i = ρ s j s i = i S + jprob i m 4 ψ 2. 6 The incompressible part of the superfluid flow velocity must be curl free since it is the gradient of a scalar, the phase of the wavefunction. Inserting Eq. into the Schrödinger equation for the Bose condensate Eq. 2, it is expressed in terms of S and the real part takes the form of the Hamilton-Jacobi equation of motion S 2 + µ = t S ρs. 7 2m 4 2m 4 ρs The last term in 7 vanishes in the incompressible limit. 3 Using this fact when taking the gradient of Eq. 7 gives 3 It is not essential to assume Planck s constant is small to be able to neglect this term since it depends only on ρ s whose gradient vanishes in the imcompressible limit.

3 3 the hydrodynamic flow equation known as Euler s equation for the superfluid part of Helium II t v s i + v s j j v s i = m 4 i µ, 8 with a hydrostatic force due to a gradient in the chemical potential. 4 What is µ? The Gibbs free energy is G = Nµ provided Ndµ = ΣdT + V dp. 9 That is, the Gibbs free energy is the product of the number of 4 He particles and the chemical potential, and this is expected since the chemical potential is the amount of energy it takes to add a single particle to the system. Denoting total density as ρ Nm4 V and the total entropy per unit mass as σ Σ Nm 4, then using Eq. 9 the gradient of the chemical potential becomes i µ = m 4 ρ ip m 4 σ i T. 0 The twofluid Navier-Stokes equation for Helium II must have viscous damping arising only from the normal flow. Furthermore, the pressure must be linear in the twofluid density and this drives first sound, p = c 2 ρ. The twofluid Navier-Stokes equation is then ρ n t v n i + v n j j v n i + ρ s t v s i + v s j j v s i = i p + η 2 v n i. With this equation known, as well as the superfluid equation, then the hydrodynamic equation for the normalfluid can be determined. The Navier-Stokes equation for the normalfluid has a nonideal equation of state but otherwise is quite standard in form. Therefore, a model of the normalfluid equation is possible with an appropriately chosen form for an interparticle potential force. III. LATTICE-BASED PARADIGM FOR QUANTUM COMPUTING Consider dynamics at the nanoscale is governed by the Schrödinger wave equation; the Hamiltonian, H, is hermitian, the evolution operator Û = e iĥt/ is unitary, and hence its quantum evolution is invertible. Consider further a nanoscale quantum device undergoing reversible evolution. For any reversible computation, one can describe the algorithm by conditional permutations corresponding to Û. For any reversible algorithm chosen, the task is to map the computational Hamiltonian of the algorithm onto the physical Hamiltonian of the nanoscale device in question. 4 The identity, 2 v2 s = v s v s+ v s v s, for the calculation of the gradient is used. Lattice based nanoscale computing offers unprecedented classical parallelism. Having information stored at atomic scales allows us to contemplate densities so high that any computation would necessarily have to be local, involving only nearby neighbors, and consequently would be ultimately fine-grained. So the issue of parallelism here involves coming up with a reasonable strategy of clocking such a large collection of qubits to achieve the computational dynamics. To this end, the lattice gas paradigm is well suited. The lattice gas paradigm is a simple spatial and temporal discretization of particle dynamics where local collisional scattering, particle translation, and nonlocal interparticle interaction are all separate and distinct computational events. Given a system with N qubits, there are 2 N basis kets in the number representation. The number of kets in what is termed the p-particle sector is equal to the binomial coefficient N choose p. 5 Suppose the quantum lattice gas wavefunction is constrained to reside in the -particle sector. The number of basis kets in this subspace of the Hilbert manifold identically equals the number of qubits since N choose = N. That is, in the -particle sector of the quantum Hilbert space, there are N amplitudes, each a complex number. So the -particle sector of an N-qubit quantum computer can be represented on a classical computer with N complex numbers. This is exactly what is done in this paper. 6 In D spatial dimensions, each site of the lattice has B nearest neighbors, the lattice coordination number. Complex amplitudes denoted by ψ,..., ψ B reside at each site of the lattice, where the position of the lattice site is specified by the vector x i, for i =,..., D. At each site, a configuration is specified by the ket ψ ψx, t =. ψ B. 2 The evolution matrix, denoted Û = ˆ + Ĵ, is unitary and collisionally scatters the incoming configuration, ψx, t, to an outgoing configuration, ψ x, t. The outgoing complex amplitudes are then streamed along ê = δ x to the neighboring lattice sites completing one time step. The quantum lattice gas microscopic trans- 5 This is because of the Pascal triangle identity P «N N p=0 = p 2 N. 6 While a classsical computer can only simulate the one-body problem using N complex amplitudes, a quantum computer in principle can simulate the full N-body problem using N qubits because of the exponential size of its Hilbert space. This clearly displays the advantage offered by a quantum computer. Yet even in the -particle sector, a fine-grained quantum computer is extremely useful since it could simulate, for example, the nonrelativistic Schrödinger equation.

4 4 port equation is 7 ψx i + εe i, t + ε 2 = ψx i, t + Ĵ ψx i, t, 3 where the lattice directions are denoted by the vectors ê. The case of one-dimension is treated here for simplicity; there are two complex amplitudes per site, ψ and ψ 2, corresponding to the positive and negative directions respectively, i.e. ψ = ψ, ψ 2. Taylor expanding the L.H.S. of Eq. 3 in space gives the associated local difference equation in time, to second order in ε, Ĵ ψt = ψt + ε 2 εĉ ψt + x + ε2 2 Ĉ2 x 2 ψt + ε The matrix Ĉ is diagonal with components of the lattice vectors of the discrete space: C αβ i = δ αβ e αi or in -dimension C =. In Eq. 4, the wavefunction s dependence on the spatial variable x is implied, and omitted here for brevity. Substituting the wave amplitude ψ = ψ 0 + ε ψ + Oε 2, 5 expanded in ε, into Eq. 4 and equating terms of similar order in ε gives the following equations 8 Ĵ ψ t = Ĉ x ψ 0 t + ε 2 6 ψ 0 t + ε 2 ψ 0 t = ε 2 Ĉ x ψ t + ε 2 ε2 2 Ĉ2 2 x ψ 0 t + ε 2. 7 Eq. 6 can be inverted to solve for the first order correction to the wavefunction, ψ. Substituting this into Eq. 7, a difference equation for ψ 0 emerges ψ 0 t + ε 2 ψ 0 t = ĈĴ Ĉ 2 x ψ 0 t + 2ε 2 + 2Ĉ2 2 x ψ 0 t + ε 2. 8 It is not possible to Taylor expand ψ 0 t + t about t because ψ is rotating. That is, the application of the unitary evolution operator + Ĵ causes large changes in the phase of the wavefunction at every time step iteration. It is possible to remedy this situation by transforming to the rotating reference frame where the wavefunction is steady. Denote the wavefunction in the rotating frame as η and the transformation matrix as X, so that 7 I am using diffusive ordering where δt δx 2 ε 2. 8 At zeroth order the operator Ĵ does not affect ψ0 since this is the equilibrium state. η X ψ. The requirement for being in the rotating frame is that the unitary evolution operator in that frame, Û = + Ĵ, be diagonal. The microscopic quantum lattice gas equation in the rotating frame becomes ηx i + εe i, t + ε 2 = Û ηx i, t, 9 where Û is diagonalized by the similarity transformation Û = ˆXÛ ˆX. Therefore, in the rotating frame Eq. 8 becomes a well defined parabolic partial differential equation in space and time t η 0 = ˆXĈ Ĵ + 2 Ĉ ˆX 2 x η 0 t. 20 Consider the following choice for the similarity transformation ˆX =. 2 2 The matrix on the right hand side of Eq. 20 must be diagonal, so Ĵ Ĉ + 2 Ĉ must be off-diagonal. This leads to the choice Ĵ = i m 2 i m, 22 which is easily inverted to give the collision matrix Ĵ = 2 i m + 2 i. 23 m 2 m That is, with these choices for the similarity transformation matrix, ˆX, and the collision matrix, Ĵ, Eq. 20 reduces to t η 0 = i 0 2m 0 x η 2 0 t. 24 Converting back to the rest frame η = ˆX ψ we have η = ψ = ψ + ψ ψ 2 2 ψ ψ 2 Therefore the first component equation of Eq. 24 becomes the Schrödinger equation of a nonrelativistic quantum particle i t Ψ = 2 2m 2 xψ, 26 where Ψ ψ +ψ 2 / 2. The density at a site, ψ +ψ 2, obeys the Schrödinger equation Eq. 26. Therefore, it is possible to recover the Schrödinger equation from a quantum lattice gas in analogy to the recovery of the Navier-Stokes equation from a classical lattice gas. 0 It is interesting to see the form the unitary evolution operator takes in both the rotating and non-rotating frames of reference. To do this it is convenient to make the following coordinate change of variables µ cos θ + i sin θ m cot θ csc θ. 27

5 Then with these definitions Ĵ = ˆXĴ ˆX = 2 i m 0 0 i m = µ 0 0 µ 28 This is easily inverted, so immediately the evolution operator in the rotating frame is found to simply be µ Û = + Ĵ = 0, 29 0 µ clearly unitary and diagonal as required. The evolution operator in the non-rotating frame is found by similarity transformation Û = ˆX Û ˆX = µ + µ µ µ cos θ i sin θ 2 µ µ µ + µ =. i sin θ cos θ 30 It is possible to generalize this approach to n-dimensions and to the many-particle Schrödinger equation. This algorithm is naturally suited for a quantum computer and can be implemented in terms of a few simple local unitary operations on a lattice of quantum bits. IV. LATTICE GAS MODEL OF HELIUM II Lattice gases with interparticle interactions have been used to model multiphase fluid, such as fluids with a liquid-gas transition. 7 2 These nonlocal lattice gas models have been extended to model fluids with multiple components. 22 This is used here to model the normalfluid part of Helium II, its mass, momentum, and entropy equations. For convenience of implementation, the lattice gas computation is done on a square lattice since the lattice Boltzmann equation for viscous fluids 23 and a quantum lattice gas for the Schrödinger equation have previously been worked out. The two fluid aspect of Helium II is modeled by coupling a multicomponent lattice Boltzmann gas to a quantum lattice gas, see figure. f 8 f " f 4 4 f 0 f 5 f 6 f 2 " 2 f 7 " 3 f " 3 FIG. : Two dimensional square lattice for Helium II lattice gas. The nine components for the normalfluid part, f 0,..., f 8, are real numbers, and four components for the superfluid part, ψ,..., ψ 4 are complex numbers. f 0 is the probability distribution for rest particles in the Boltzmann gas. TABLE I: Identifications for Multicomponent Lattice Boltzmann Gas. Lattice Boltzmann Helium II mass m m 4 Mass of He 4 atom number density f ρ n/m 4 Normalfluid number density velocity v v n Normalfluid velocity mass m 2 0 number density f 2 ρξ Effective entropy density velocity v 2 v n Normalfluid velocity Both the classical lattice gas and the quantum lattice gas have appropriately chosen potentials to provide the correct coupling between the normalfluid and superfluid species; in the former this is a long-range interparticle potential and in the latter this is the chemical potential. To model the normalfluid part of Helium II, a twocomponent lattice Boltzmann gas is employed. With a particular choice of an interparticle interaction, the first component represents the motion of the normalfluid. A local momentum change has the form p/τ = i V. In this way the normalfluid suffers an external potential force as the gradient of a scalar. The second component represents a passive scalar field, the entropy density of the system. The second component has no interparticle interaction and is advected by the normalfluid. However, gradients in the passive scalar field affect the flow because the equation of state for the fluid depends on this passive scalar. Table I lists the lattice Boltzmann quantities in correspondence to like quantities in Helium II. For small Mach number and in the incompressible regime, the continuity equation, the Navier-Stokes equation for the normalfluid velocity, and the effective entropy density equation, ρξ, advected by the normalfluid are the following t ρ n + i ρ n v n i = 0 3 t v n i + v n j j v n i = i p LB i V + ν 2 v n i 32 t ρξ + i ρξv n i = D 2 ρξ, 33 where the pressure, kinematic viscosity, and diffusion coefficient are respectively p LB c 2 sρ n ν = 3 τ 2 D = 3 5 τ 2, 2 34 where c s is the sound speed, and τ and τ 2 are the relaxation times in the lattice Boltzmann BGK collision operators. 22 Consider a situation where the lattice gas system is at rest with constant background density and where the passive scalar field is constant. When the system is subjected to a small perturbation, the macroscopic dynamical variables can be ε expanded. Then considering only first order fluctuations in the macroscopic vari-

6 6 ables, the lattice gas linear hydrodynamics regime is obtained by making the following expansions vi s = εu s i vi n = εu n i 35 ρ s = ρ s + εϱ s ρ n = + εϱ n 36 ξ = ξ + ε ξ. 37 Choose the interparticle potential, V = V ρ s, ρ n, ξ, to have the following form 9 V = p LB + ρc 2 s + ρ s ξ, 38 ρ where ρ + ρ s. Expression 38 is the correct form of the interparticle potential because it gives rise to superfluid-like hydrodynamic equations for the coupled lattice gas systems. Then, the Navier-Stokes equation for the normalfluid part becomes t v n i + v n j j v n i = ρ i p ρ s i ξ + ν 2 v n i, 39 where the twofluid pressure is p = c 2 sρ. The density, ρ s, and velocity, vi s, of the quantum lattice gas are defined in terms of the amplitude, Ψ = a ψ a, as follows Re{Ψ} ρ s = Re{Ψ} 2 +Im{Ψ} 2 vi s = i arctan. Im{Ψ} 40 Choose the chemical potential to be the following 0 µ = ρ ρ c 2 s ξ, 4 which is the external potential in the Schrödinger equation. Conservation of probability implies a mass continuity equation for the superfluid density. So the macroscopic continuity equation and Euler s equation for the superfluid velocity Eq. 8 are t ρ s + i ρ s v s i = 0 42 t v s i + v s j j v s i = ρ i p QLG + i ξ, 43 where p QLG c n 2 ρ and m 4 is taken to be unity in the numerical simulation note that c n is the sound speed of first sound. Note that the interparticle potential, V, 9 An alternate, but equivalent, form of the interparticle potential is V = p n ln ρ ρ n + ρs ξ, where p n = c 2 s. 0 An alternate, but equivalent, form of the chemical potential is µ = c 2 s ln ρ ξ. and the chemical potential, µ, were chosen so that the following two conditions are satisfied p = p LB + ρ s ρ p QLG p LB = p QLG ρ. 44 The first condition is necessary to recover first sound in the numerical simulation. It enforces the requirement that density waves propagate at the same speed in both the normalfluid and superfluid parts. The second condition is necessary for second sound, where relative motion within the twofluid system causes a cancellation of any pressure variation as entropy waves propagate. V. LATTICE GAS HYDRODYNAMICS The combined lattice gas mass continuity equation is t ρ s + ρ n = i ρ s v s i + ρ n v n i. 45 The combined lattice gas twofluid flow equation is ρ n t v n i +v n j j v n i +ρ s t v s i +v s j j v s i = c n 2 i ρ+η 2 v n i. 46 The quantum lattice gas superfluid flow equation is t v s i + v s j j v s i = c2 s ρ iρ + i ξ. 47 The classical lattice gas normalfluid flow equation is t v n i + v n j j v n i The passive scalar equation is = c2 s ρ iρ ρ s ρ n i ξ + ν 2 v n i. 48 t ρξ + i ρξv n i = D 2 ρξ. 49 Using 35, 36, and 37 and neglecting damping, the linearized hydrodynamic lattice gas equations are the following t ϱ = i ρ s u s i + u n i + Oε 2 50 t u s i t u n i = c2 s ρ i ϱ + i ξ + Oε 2 5 = c2 s ρ i ϱ ρ s i ξ + Oε 2 52 ρ t ξ + ξ t ϱ = ξ i ρ s u n i + u n i + Oε 2 53 Consider a situation where the lattice gas has a constant passive scalar field, so ξ = 0. The fluctuating part of the twofluid mass density is ϱ = ρ s + ρ n, and the fluctuating part of the twofluid current density is j i = ρ s u s i + u n i. So the linearized twofluid equation, sum of Eq. 5 and Eq. 52, and the linearized mass continuity equation Eq. 50 reduce to t ϱ = i j i t j i = c 2 s i ϱ. 54

7 7 Eliminating j i directly gives a wave equation for the density of the twofluid lattice gas system 2 t ϱ = c 2 s 2 ϱ. 55 The speed of first sound in the numerical simulation is simply the usual lattice gas sound speed c = c s. Next, consider a situation where the lattice gas is kept at a fixed pressure, so p = 0. So the linearized superfluid Eq. 5 and the linearized normalfluid Eq. 52 reduce to t u s i = i ξ t u n i = ρ s i ξ. 56 Subtracting these gives an equation for fluctuation of the velocity difference u n i us i t u n i u s i = ρ i ξ. 57 So in this model, a gradient in the passive scalar field produces relative motion between the normalfluid and superfluid parts of the system. Next, to obtain another equation for relative motion, insert the linearized mass continuity Eq. 50 in the linearized passive scalar Eq. 53 which becomes ξ i u n i u s i = ρ ρ s t ξ. 58 Now we have two equations from which the relative motion u n i us i can be eliminated to give a wave equation for the passive scalar 2 t ξ = c ξ, 59 where the speed of second sound in the lattice gas is c 2 ρs ξ, a second kind of hydrodynamic sound wave in the numerical simulation analogous to that which occurs in Helium II, see figure 2. A square lattice grid is used in the simulation. To initialize the two-dimensional numerical simulation, background densities, ρ s,, and ξ, are chosen. Then a sinuoidal density perturbation in the passive scale field is made with a maximum size wavelength equal to the size of the spatial lattice, ξx, t = sin2π x 52. The flow velocities are zero. The fluctuation in time of the passive scalar field, δξt, is measured by integrating over one of the lattice directions as follows: δξt 52 dx ξx, t ξx, t. This integration is done to improve the statistics so as to remove noise. The result is the waveform depicted in figure 2 which is caused by 59. VI. CONCLUSION We see that an array of complex amplitudes undergoing a unitary and local evolution of the type proposed above acts like a superfluid. Some thoughts developed from observations of the coupled lattice gas model of the Helium II quantum fluid are presented for future consideration. Entropy Fluctuation Time Step FIG. 2: Second sound wave. Sinusoidal entropy fluctuation due to an initial perturbation in the entropy field with an initial constant density in the normalfluid and superfluid parts. Graph of the oscillation of the entropy field. The density is probabilistic since ρ m 4 ψ 2 requires a course-grained measurement of the wavefunction. The size and duration of the course-graining is not prescribed, but because the measurement must occur, the classical lattice gas functions as an environment that causes the quantum state to decohere. Since the measurement of the density may be done over a coarse-grained block, it need not completely destroy the entangled states of the qubits. The coupled system, a classical lattice gas in contact with a quantum lattice gas, will have inherent fluctuations induced by these periodic measurements. The quantum wavefunction will be coherent only for short times since it is necessary to determine the classical interparticle potential, V, by collapsing the wavefunction periodically in time step fashion. Therefore, to build a quantum computer to simulate finite-temperature Helium II, only short coherence times are required. So the burden of avoiding decoherence of the quantum computer s wavefunction is alleviated since the wavefuction is already periodically collapsed by its contact with the classical lattice gas system. Controlled decoherence is a preplanned design feature of the quantum computer. Finally, consider the situation where one might attempt to model superfuild Helium II at zero temperature. In this situation, the classical lattice gas would not be implemented to model the normal part of the fluid. Yet, any actual quantum computer would necessarily run at finite temperature and so thermal excitations will be present and decoherence of the quantum computer s wavefunction would necessarily remain in a uncontrolled fashion. It is possible that quasiparticles could exist and represent a weakly interacting gas in a fashion perhaps analogous to that which actually occurs in Helium II at finite temperature. Therefore, the theoretical analysis given above is relevant to a quantum lattice gas even if no coupling is made to a classical lattice gas. This is an issue worthy of further exploration.

8 8 VII. ACKNOWLEDGEMENTS I would like to thank Bruce Boghosian, Xiaowen Shan, Francis Alexander, and Norman Margolus for their help and constructive discussions. Computation was performed on the SGI Power Challenge Array at Center for Comp. Sci. at Boston University. This work is supported under initiative No. 2304CP by the Mathematical and Computatoinal Sciences Directorate of the Air Force Office of Scientific Research at Bolling AFB, Washington D.C. Richard P. Feynman. Simulating physics with computers. International Journal of Theoretical Physics, 26/7: , Edward Fredkin and Tommaso Toffoli. Conservative logic. International Journal of Theoretical Physics, 23/4:29 253, Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DeVincenzo, Norman H. Margolus, Peter W. Shor, Tycho Sleator, John Smolin, and Harald Weinfurter. Elementary gates for quantum computation. Physical Review A, 525: , A.R. Calderbank and Peter W. Shor. Good quantum error correcting codes exist. LANL archive: quant-ph/952032, Charles H. Bennett. Quantum information and computation. Physics Today, October:24 30, Artur Ekert and Richard Jozsa. Quantum computation and shor s factoring algorithm. Reviews of Modern Physics, 683: , Seth Lloyd. Quantum-mechanical computers. Physics Today, Oct:40 45, Peter W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. In Proceedings of the 35th Annual Symposium on Foundations of Computer Science, pages Santa Fe, NM, IEEE Computer Society Press, David G. Cory, Amr F. Fahmy, and Timothy F. Havel. Ensemble quantum computing by nuclear magnetic resonance spectroscopy. Technical report tr-0-96, Harvard University Center for Research in Computing Technology, Aiken Computation Laboratory, 33 Oxford Street, Cambridge, MA 0238, December Uriel Frisch, Brosl Hasslacher, and Yves Pomeau. Latticegas automata for the navier-stokes equation. Physical Review Letters, 564: , 986. Bruce M. Boghosian and Washington Taylor. A quantum lattice gas models for the many-body Schroedinger equation. LANL Archive, quant-ph/ , Sauro Succi. Numerical solution of the Schroedinger equation using discrete kinetic theory. Unpublished, S. Succi and R. Benzi. Lattice boltzmann equation for quantum mechanics. Physica D: Nonlinear Phenomena, 693-4: , I.M. Khalatnikow. An Introduction to the Theory of Superfluidity. Addison-Wesley Publishing Company, Inc., 2nd edition, 989. Originally published in 965 as part of the Frontiers in Physics Series. 5 David R. Tilley and John Tilley. Superfluidity and Superconductivity. Graduate Student Series in Physics. Adam Hilger Ltd, Bristol and Boston, 2nd edition, 986. Published in association with the University of Sussex Press. 6 L.D. Landau. The theory of superfluidity of helium ii. Journal of Physics, V:7 90, Xiaowen Shan and Hudong Chen. Lattice Boltzmann model for simulating flows with multiple phases and components. Physical Review E, 473:85 89, Xiaowen Shan and Hudong Chen. Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation. Physical Review E, 494: , Jeffrey Yepez. Lattice-gas crystallization. Journal of Statistical Physics, 8/2: , Jeffrey Yepez. Long-range lattice-gas simulation on the CAM-8 prototype. Technical report pl-tr , Phillips Laboratory, PL/GPA Hanscom AFB, MA 073, September Jeffrey Yepez. A lattice-gas with long-range interactions coupled to a heat bath. American Mathematical Society, 6:26 274, 996. Fields Institute Communications; Presented at the 993 Lattice-Gas Conference held in Waterloo, Canada. 22 Xiaowen Shan and Gary Doolen. Multi-component lattice- Boltzmann model with interparticle interaction. Journal of Statistical Physics, 8:379, Danial O. Martinez, William H. Matthaeus, Shiyi Chen, and Dave Montgomery. Comparison of spectral method and lattice bolzmann simulations of two-dimensional hydrodynamics. Physics of Fluids, 63: , 994.