Quantum Computation Of Fluid Dynamics

a large coherent quantum computer is difficult, if not all together impossible,and predicting its behavior by analytical means is intractable.So what can be done about this? What I would like to consider is a simplification that will sidestep these obstacles and give us two important advantages:(1) a simple way to use a quantum computer with a large number of qubits;and (2) a way to analysis of its behavior. In lattice-gas quantum computationcomplete coherence of the wavefunction is not needed for the algorithm to work.In fact, we assume entanglement of qubit states is only among small clusters ofqubits in a localized nearby neighborhood, so independent quantum operationsare done in a classically parallel fashion on all sites simultaneously. This is thecollision step.In a deterministic classical lattice gas, the collision operator is a permutationmatrix with components being either zero or one. In a probabilistic classicallattice gas, the collision operator is a transition matrix with real valued components. In contrast, in a quantum lattice gas, the collision operator can be aunitary matrix with complex components. The collision process is in generalirreversible because a projection of the quantum computer’s wavefunction intoa tensor product state over the qubits is periodically made causing the wavefunction to partially collapse. Hence, application of Ŝ does not cause any globalentanglement.The quantum lattice gas presented here should not be confused with previous quantum lattice gas models by Succi [17], Boghosian [18], or Yepez [19] forsimulating quantum mechanical systems. Despite some similarities, the type ofquantum lattice gas treated in this paper is a direct generalization of a classical lattice gas with quantum bits replacing classical bits. In fact, if orthogonalpermutation matrices with 0 and 1 components are used for the collision process, in the limit of complete collapse of the lattice-gas quantum computer’swavefunction, the quantum lattice gas exactly reduces to a classical lattice gas.This particular feature distinguishes the quantum lattice gas for fluid simulationfrom the quantum lattice gases for quantum mechanical simulation.4PreliminariesConsider a lattice-gas quantum computer with the following properties: V is the number of lattice sites B is the number of qubits per site (and the number of nearest neighbors) N V B is the total number of qubits 2N is the size of the full Hilbert space 2B is the size of the on-site submanifold, denoted H B is the size of the reduced on-site submanifold, denoted BWe will use the following convention for indices:5

Constants τmcDBeaiai, j, k, lNameslength unittime unitmass unitvelocity unit ( τ )spatial dimensionlattice coordination numberunit lattice vectorsdirectional index (1,2,. . . , B)spatial indicesTable 1: Model ConstantsTable 2: Wavefunction SymbolsSymbolΨψωqSize of Manifold2N2BB2DescriptionTotal system wavefunctionOn-site ketPartially collapsed on-site amplitudesQubit ket Small roman letters (a, b, c) for the B-space dimensions, a {1, . . . , B} Greek letters (α, β, γ) for the H-space dimensions, α {0, . . . , 2B 1} Middle roman letters (i, j, k) for the spatial dimensions, i {1, . . . , D}The full Hilbert space of size 2BV is partitioned into V independent quantummanifolds of dimension 2B , as depicted in Fig.1. Quantum superposition ofstates occurs only within each 2B -dimensional subspace, denoted H. A generalon-site ket defined over the basis states of H is the following ψ0B2X 1 ψ1 ψα ( x, t) αi (5) ψ( x, t)i . .α 0ψ2B 1The ket ψi is specified by 2B complex amplitudes, denoted ψ0 , . . . , ψ2B 1 . Thequantum computer’s total wavefunction is formed as a tensor product over allthe H-manifoldsVO ψ( x, t)i.(6) Ψ( x1 , . . . , xV ; t)i x 1The collision operator, Ĉ, is blocked over all the H-manifolds. That is, thecollision matrix is block diagonal with V blocks each of size 2B 2B , and6

Lattice NetworkSingle Lattice Site2Bψ0ψ1ψ2ψ3Dimensionsψ62ψ63 000000 000001 000010 000011 111110 111111 Figure 1:An array of small quantum computers (the quantum computers are depicted ascircles) arranged in a 2-dimensional triangular lattice (B 6). The large circle on the right isan expanded view of a single quantum computer, which is one site of the lattice. It depictsthe on-site submanifold, H. Each quantum computer at a lattice node has 6 qubits so theon-site ket ψi resides in a 64-dimensional Hilbert space. Each node is coupled to its 6 nearestneighboring quantum computers by a mechanism allowing for the exchange of a single qubit.therefore can be written as a tensor productĈ VOÛ .(7)x 1The on-site collision matrix, Û , is unitary and acts on the on-site ket ψ 0 ( x, t)i Û ψ( x, t)i.(8)The prime on the L.H.S. of (8) indicates that the ket is an outgoing collisionalstate.5Unitary Collision MatrixLet Q̂α be matrices representing the conserved quantities in the single speedquantum lattice gas, Q̂α (Q̂ , Q̂i ) where i is an index over the independentspatial coordinates. A fundamental property of a quantum lattice gas is thatthe mass density and the momentum density can be written as followsρ hψ Q̂ ψiρvi hψ Q̂i ψi,(9)(10)that is, where the component of Q̂ are(Q̂ )µν mδµνBXa 17bµa ,(11)

and where the component of Q̂i are(Q̂i )µν mcδµνBXbµa eai .(12)a 1The bµa denotes the ath -bit of the µth ket. The êa here denote the unit latticevectors where a 1, . . . , B. The components of the 2B 2B qubit numberoperator are defined by(13)(n̂a )µν bµa δµν .In terms of (13), the operators for mass and momentum areQ̂ mBXn̂a(14)eai n̂a .(15)a 1andQ̂i mcBXa 1In terms of (13) the invariant quantities are simply expressed as the followingmatrix elementsρ BXmhψ n̂a ψi(16)mceai hψ n̂a ψi.(17)a 1ρvi BXa 1The matrix elements (9) and (10) must remain constant after each time stepiteration(18)hψ(t τ ) Q̂α ψ(t τ )i hψ(t) Q̂α ψ(t)i.Since ψ(t τ )i Û ψ(t)i, this implies thatÛ † Q̂α Û Q̂α ,(19)[Û , Q̂α ] 0.(20)which is just the commutatorThe matrices Q̂α must commute with Û .Let ĝ denote the generator of ÛÛ eiεĝ ,(21)where ε is an “Euler angle”. Consider a “rotation” through an infinitesimalangle ε so that Û can be Taylor expanded to first order asÛ 1 iεĝ.8(22)

The unitary condition, Û † Û 1, implies that the generator is hermitianĝ ĝ † 0 O(ε2 ).(23)From (19), we see that mass and momentum conservation is ensured providedQ̂α ĝ ĝ † Q̂α 0 O(ε2 ).(24)The solution of the set of linear equations (23) and (24) give the Lie algebra forthe unitary group. Therefore, the mass density (9) and the momentum density(10) are conserved when each equivalence class block of the collision operator isan element of the unitary group U (n) where n is the size of the equivalence classof the incoming local configuration. This is an important feature of a quantumlattice gas. Since any member of the unitary group can be used, the quantumlattice gas is algorithmically robust.Equivalence Class:m 2 p 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 Equivalence Class:m 3 p 0 1 0 1 0 1 0 0 1 0 1 0 1 Figure 2:The equivalence classes for the quantum FHP lattice gas.An equivalence class is defined as a set of basis states that correspond to particle configurations with the same mass and momentum. The unitary collisionoperator, Û , acting on the 2B dimensional H-manifold itself is block diagonalover the equivalence classes. For example, there are two equivalence classes forthe FHP lattice gas [2], see Fig.2. The first equivalence class is comprised of thefollowing two-body kets 9i 001001i 18i 010010i 36i 100100i9

with mass, m 2, and zero momentum, p 0. A general ket in this massmomentum sector of the on-site manifold is a linear combination of theseα 100100i β 010010i γ 001001i,(25)where α, β, and γ are complex numbers. The second equivalence class is comprised of the following three-body kets 21i 010101i 42i 101010iwith mass, m 3, and zero momentum, p 0. A general ket in this massmomentum sector is a linear combination of theseµ 101010i ν 010101i.(26)So Û for a two-dimensional quantum lattice gas on a triangular lattice has twoblocks, a U (3) block for mixing the 2-body configurations and a U (2) block formixing the 3-body configurations.For the triangular quantum lattice gas, we have iζ 0 ψ21eiξ sin ηe cos ηψ21iθ e,(27)0ψ42ψ42 e iξ sin η e iζ cos ηwhere zero momentum three-body configurations are mixed by a unitary matrix,U (2) U (1) SU (2), which in general has four free parameters. The zeromomentum two-body configurations are mixed by a unitary matrix, U (3) U (1) SU (3), which in general has nine free paramters8 0 ψ9ψ90 ψ18 eiθ SU (3) ψ18 .(28)0ψ36ψ366Partial Collapse of Post-Collision Ket ψ 0 iTo avoid causing any global entanglement as induced by streaming, we projectthe post-collision ket ψ 0 i which resides in the 2B -dimensional H manifoldonto a smaller B-dimensional submanifold, B. This is done using a projectionoperator, denoted Γ̂, as follows ω0 1 ω20 ω 0 i Γ̂ ψ 0 i . .0ωB(29)The operator Γ̂ causes a partial collapse of the locally entangled on-site state ψiresulting in a nonentangled state ωi residing in a smaller manifold (a mapping8 Wedo not write out the SU(3) matrix in component form because it is too complicated.10

2BBDimensionsψ0ψ1ψ2ψ3 000000 000001 000010 000011 ψ62ψ63 111110 111111 Dimensionsω2ProjectionInjectionω 1 1 ω 2 2 ω 3 3 ω 3ω 4 4 ω 5 5 ω 6 6 ω1ω6ω4ω5Figure 3:Two-dimensional quantum lattice gas on a triangular lattice. The lattice coordination number is B 6. There are 26 64 amplitudes in the H-manifold and 6 amplitudes inthe B-manifold. Injection maps the 6 on-site amplitudes ωa in the B-manifold into the largerH-manifold. The inverse process, projection, maps the 64 amplitudes ψα in the H-manifoldonto 6 amplitudes ωa in the B-manifold. The projection is a measurement process that causesa partial collapse of the on-site wavefunction ψ( x, t)i. The partially collapsed wavefunctionis ω( x, t)i.from 64 dimensions down to 6, see Fig.3). Thus ωi in (29) may be termed thecollapsed post-collisional on-site ket. By construction, the action of Γ̂ fixes thephase, θa , of the on-site qubits qa i according to the following recipe qa i cos θa 1i sin θa 0i,where ωa cos θa . That is, qa i ωa 1i p1 ωa ωa 0i.(30)(31)After the collapse of the ket ψi the single-particle occupation probability, denoted fa , is a well-defined quantity. It is the probability of finding a particle atcoordinate ( x, t) with momentum mcêafa ( x, t) ωa ( x, t)ωa ( x, t).(32) 1 0Using the single qubit number operator n̂ , (32) can be written in0 0terms of the qubit ket asfa ( x, t) hqa ( x, t) n̂ qa ( x, t)i.(33)Furthermore, in (16) and (17), the matrix element hψ n̂a ψi also gives theprobability of particle occupancy. So for a quantum lattice gas, fa can also beexpressed as the matrix element of the multiple qubit number operator9fa hψ n̂a ψi.9 In(34)a classical lattice gas the single-particle occupation probability is obtained by ensembleaveraging over the number variables, fa hna i, where na 0 or 1.11

Inserting the expression for the outgoing collisional state (8) into the R.H.S.of (29), we have(35) ω 0 i Γ̂Û ψi.In §7 we will use a nonlinear function for the projection operator. Our goalis to retain as much quantum information as possible while allowing (35) toreduce to the collision equation of classical lattice gas transport when Û is areal-valued permutation matrix. This is accomplished by projecting down fromthe H-manifold containing 2B complex amplitudes to the B-manifold with onlyB complex amplitudes. Each ωa (or associated qubit qa i) is “attached” toone of the lattice directions. The reason for this reduction of the quantuminformation is the following. By using only B complex amplitudes, one foreach direction, we can straightforwardly write down a quasi-classical streamingequation in analogy to the streaming equation of a classical lattice gas ω( x L̂a , t τ )i ω 0 ( x, t)i,(36)where Labi êai δab . Inserting (35) into (36), we have ω( x L̂a , t τ )i Γ̂Û ψ( x, t)i.(37)The only task remaining to complete the analogy to classical lattice gas dynamics is to rewrite the R.H.S. of (37) solely in terms of the ω’s. This can bedone by injecting the on-site collapsed ket ωi residing in the sub-manifold Bup into the larger on-site manifold H (see Fig.3). This process can be expressedˆ as followsby application of an injection operator, I, ψ( x, t)i Iˆ ω( x, t)i.(38)A straightforward way to accomplish the injection is to take the tensor productover the on-site qubits ψ( x, t)i BO qa ( x, t)i.(39)a 1This is a nonlinear operation.10 Let us revisit the example a two-dimensionalquantum lattice gas on a triangular lattice (B 6), a generalization of theclassical FHP lattice gas [2]. Fig.3 illustrates projection from the 64-dimensionalH-manifold down to the 6-dimensional B-manifold and illustrates injection fromthe B-manifold up to the H-manifold. The 6 on-site amplitudes ωa (or theassociated 6 on-site qubits qa i) generated by the projection can be streamedin a classical fashion. After streaming to their new sites, each quantum computerhas a new incoming configuration of the ωa amplitudes. Before this configurationcan be collided, they must be injected up to the larger 64-dimensional manifoldwhere the collision process is well defined.10 A non-square linear matrix could also be used, but it is difficult to find an appropriatematrix even though it can be shown that one exists.12

Inserting (38) into (35) gives ω 0 ( x, t)i Γ̂Û Iˆ ω( x, t)i.(40)Using the fact that the projection of the injection is the identity operation: ωi Γ̂Iˆ ωi, we write (40) in a form analogous to the classical lattice gascollision equationhi(41) ω 0 ( x, t)i ω( x, t)i Γ̂Û Iˆ ω( x, t)i Γ̂Iˆ ω( x, t)i .Finally, we arrive at the quantum lattice gas microscopic transport equation byinserting (41) into (36) ω( x L̂a , t τ )i ω( x, t)i Ω( x, t)i,(42)where the quantum lattice gas collision operator is defined as Ω( x, t)i Γ̂Û Iˆ ω( x, t)i Γ̂Iˆ ω( x, t)i.(43)In component form, (42) isωa ( x êa , t τ ) ωa ( x, t) Ωa (ω ( x, t)) .(44)Equation (44) is identical in form to the classical lattice gas transport equationwhere the occupation variable, na 0 or 1, is replaced by a complex amplitude,0 ωa 1, that continuously encodes the square root of the probability forparticle occupancy. Hence (44) is a much more useful expression of the quantumlattice gas dynamics than (4) is.7The Projection OperatorWe can write an analytical expression for the projection operator where theamplitudes ωa a nonlinear function of the amplitudes ψavu2B 1uXωa Γ̂a (ψ) t ψα 2 bαa .(45)α 0Note that bαa 0 or 1 is the Boolean value of the ath bit of the αth ket in thenumber representation. Let m̂ and p̂i be the operators for mass and momentumin the B-space. Then the matrix element for the mass density isρ hω m̂ ωi,(46)where m̂ab mδab , and the matrix element for the momentum density isρvi hω p̂i ωi,13(47)

where (p̂i )ab mceai δab . Q̂ and Q̂i were defined in §5 to be the mass andmomentum operators in ψ-space. Here we have mass and momentum operatorsin ω-space. The matrix element (9) defines the mass density as ρ hψ Q̂ PBψi, where (Q̂ )αβ m a 1 baα δαβ , and the matrix element (10) defines thePBmomentum density as ρvi hψ Q̂i ψi, where (Q̂i )αβ mc a 1 baα eai δab .Equating (46) with (9) and equating (47) with (10) gives us a way to check theprojection operator (45).This is done as followshω m̂ ωi mBX ωa 2(48)a 1B m2B XX ψα 2 bαaa 1 α 1B 2X2 ψα mα 1BX(49)!bαa,(50)a 1where we used the square of (45) on the second line of the derivation. Therefore,we have(51)hω m̂ ωi hψ Q̂ ψi,where the mass operator in ψ-space(Q̂ )αβ mBXbαa δαβ(52)a 1is identical to Q̂ defined in (9). So the projection operator (45) conserves mass.We continue the consistency check by rewriting (47)hω p̂i ωi mcBX ωa 2 eai(53)a 1B mc2B XX ψα 2 bαa eaia 1 α 1B 2X2 ψα α 1mcBX(54)!bαa eai,(55)a 1where again we used the square of (45) on the second line of the derivation.Therefore, we have(56)hω p̂i ωi hψ Q̂i ψi,where the momentum operator in ψ-space(Q̂i )αβ mcBXa 114bαa eai δαβ ,(57)

is identical to Q̂i defined in (10). So the projection operator (45) conservesmomentum as well as mass.8Equilibrium AnsatzThe ωa amplitudes in B-space can be ordered in powers of ε as follows ωi ω (0) i ε ω (1) i O(ε2 ),(58)where ω (0) i denotes the equilibrium ket and where the ket ω (1) i is the firstorder correction from equilibrium. The condition equilibrium is that ω (0) isatisfies the following identity ω (0) i Γ̂Û Iˆ ω (0) i.(59)Note that the associated equilibrium ket in H-space, ψ (0) i, follows from (58)by injection ψi Iˆ ωi ψi ψ (0) i ε ψ (1) i O(ε2 ).(60)So we can also write (59) as follows ψ (0) i Û ψ (0) i.(61)It is clear that ψ (0) i is an eigenvector of Û with unity eigenvalue. Using (59),we immediately see that the collision operator (43) vanishes at equilibrium Ω(ω (0) )i Γ̂Û Iˆ ω (0) i Γ̂Iˆ ω (0) i 0.(62)Equations (58) and (59) constitute the essential ansatz that will allow us toperform a Chapman-Enskog analysis of the quantum lattice gas. It is possible toanalytically solve (59) for ω (0) i. Knowing the form of ω (0) i, we can predict thehydrodynamics equations of the quantum lattice gas at the macroscopic scale.In the Chapman-Enskog analysis, we expand the collision operator, Ωi, aboutthis equilibrium ket ω (0) i. In so doing, the Jacobian of the collision operatoris computed as a first order correction and is evaluated at ωi ω (0) i. Thetransport coefficients for the mass diffusion, shear viscosity, and bulk viscositydepend on the value of this first order correction, and this in turn depends onthe value of ω (0) i. Hence, one must determine the equilibrium amplitudes inorder to compute the value of the transport coefficients. (0) (0)(0)The a single particle occupancy probability fa ωa ωa hqa n̂ (0)qa i hψ n̂a ψi has the functional formfa 1,eαρ βêa · p γE 1(63)where the argument of the exponential is a linear combination of the conservedscalar quantities: (1) the mass ρ; (2) the momentum component êa · p along15

(a)(b)Single-Particle DistributionSingle-Particle Distribution0.450.3f60.25f10.20.15f2f20.05 0.1 0.15 0.2 0.25 0.3Velocityf1f60.4f60.35f10.30.25f20.20.150.05 0.1 0.15 0.2 0.25 0.3VelocityQuantum lattice gasClassical lattice gasVelocityFigure 4:Theo

quantum particle system is termed a quantum lattice gas and the associated quantum computer network is called a lattice-gas quantum computer. Over a decade ago, classical lattice gases were found that behave like a vis-cous Navier-Stokes fluid at the macroscopic scale [1, 2]. In this paper we show that a quantum lattice gas does too.