xi'jj

wi j fj(xj)

|R(xf,T)> ' G(xf,T;x0,0) |R(x0,0)>

' m(xf,T)

(x0,0)

D[x(t)] exp( iSm

T

0

dJ[ 12

m0x2&V(x)]) |R(x0,0)>

'limN64 m

(xN%1 xf,T)

(x0,0)

dx1@@@dxN ( m2BiS)t

)(N 1)/2exp( i)tS j

N

j 0[ m

2(xj 1&xj

)t)2 & V(xj)])|R(x0,0)>

(1)

(2)

A Quantum Dot Neural Network

E.C. Behrman and J. NiemelDepartment of Physics, Wichita State University, Wichita, KS 67260-0032

J.E. SteckDepartment of Mechanical Engineering, Wichita State University, Wichita, KS 67260-0035

S.R. SkinnerDepartment of Electrical Engineering, Wichita State University, Wichita, KS 67260-0044

Abstract

We present a mathematical implementation of a quantum mechanical artificial neural network, in the quasi-continuumregime, using the nonlinearity inherent in the real-time propagation of a quantum system coupled to its environment. Ourmodel is that of a quantum dot molecule coupled to the substrate lattice through optical phonons, and subject to a time-varying external field. Using discretized Feynman path integrals, we find that the real time evolution of the system can beput into a form which resembles the equations for the virtual neuron activation levels of an artificial neural network. Thetimeline discretization points serve as virtual neurons. We then train the network using a simple gradient descent algorithm,and find it is possible in some regions of the phase space to perform any desired classical logic gate. Because the networkis quantum mechanical we can also train purely quantum gates such as a phase shift.

I. INTRODUCTION

Many artificial neural networks are simulations, running on algorithmic computers [1]. With this, the massiveparrallel processing speed advantage of a neural network is lost. Clearly it would be better to utilize the intrinsic physicsof a physical system to perform the computation. Many efforts have been expended in this direction, using systems rangingfrom nonlinear optical materials to proteins [2]. At the same time, many other workers have been exploring the possibilityof building quantum computers [3]. In this paper we present an architecture for a quantum neural computer, using the realtime evolution of quantum dot molecules [4], and show by simulation that such an architecture can perform any classicallogic gate. Since the time evolution is quantum mechanical, it can compute backwards or forwards in time; moreover, it cancalculate a purely quantum mechanical gate, such as a phase shift, for which there is no classical equivalent.

II. MATHEMATICAL DEVELOPMENT

In most artificial neural network implementations, the neurons receive inputs from other processors via weightedconnections and calculate an output which is passed on to other neurons. The calculated output x of the i neuron isi

th

performed on the signals {x} from the other neurons in the network and is given byj

where w is the weight of the connection from the output of neuron j to neuron i, and f is a bounded differentiable real valuedij j

neuron activation function for neuron j [1]. Similarly we can write the expression for the time evolution of the quantummechanical state of a system:

|R(Fz(N)t),T)>' j{Fz(j)t)}

exp( iSjj

[KFx(j)t) % 0(j)t)Fz(j)t)] ) I [Fz(t)] |R(Fz(0),0>

2

Figure 1 A quantum dot molecule, the physicalmodel for our quantum neural network. The circlesrepresent the quantum dots, which are spatially closeenough that electrons can tunnel between neighboringdots. Coulombic repulsion between the two electronsresident in the molecule gives it two possible (ground)states, P=1 and P= -1.

(3)

Here |R(x ,0)> is the input state, the initial state of the quantum system. |R(x ,T)> is the output state, the state of the system0 f

at time t=T. G is the Green's function, which propagates the system forward in time, from initial position x at time t=0 to0

final position x at time t=T. The second line of Equation 2 expresses G in the Feynman path integral formulation of quantumf

mechanics [5], in which G is thought of as the infinite sum over all possible paths that the system could possibly take to getfrom x to x . This is indicated by the notation D[x(t)], an infinitesimal change in the path x(t). Each path is weighted by the0 f

complex exponential of the phase contributed by that path, given by the classical action for that path; m is the mass, 2BSis Planck's constant, and V is the potential energy. This is equivalent to the third line, in which the paths are discretized:N)t=T, with the number of discretization points, N, 6 4. If, instead, we take N to be finite, we have a “quasi-continuum”,and the N intermediate states can be considered to be the states of N quantum neurons, one at each time slice j)t. Thenonlinearity necessary for neural computation is inherent in the “kinetic energy” term, (x - x) , and in the exponential. j+1 j

2

Each of the N neurons’ different possible states contributes to the final measured state; the amount it contributes, can beadjusted by changing the potential energy, V(x).

III. MODEL SYSTEM

Our model system is that of a quantum dotmolecule [4, 6], with five dots arranged as the pips on aplaying card. The dots are close enough to each other thattunneling is possible between any two neighbors. Twoelectrons are fed into the molecule, which then has adoubly-degenerate ground state (in the absence ofenvironmental potentials). These are shown in Figure 1,and can be thought of as the “polarization” P of themolecule, equal to ±1, that is, the Pauli matrix operator F .zIn Equation 2 this would be the value of the positionvariable x at a given time t, x(t).

In addition to adjusting or training V(x), we canobtain an additional trainable nonlinearity by coupling thequantum system to its environment. We choose thisenvironment to be a set of Gaussians, that is, theenvironment has a quadratic Hamiltonian, or, equivalently,a normal distribution; if the set is taken to be infinite, anydesired influence including dissipation can be produced[7]. In our model this would be represented by thecoupling between the electronic state of the dot moleculesand the lattice through optical phonons. Physically the coupling would have to be weak enough to be represented accuratelyas linear; for example, GaAs substrate satisfies this, with a (unitless) electron-phonon coupling parameter of 0.08 <<1 [8]Equation 2 becomes:

where the path integral over possible positions at each time, x(t), has been written as a finite set of sums over states of thepolarization, F , at each time slice j)t: at each time slice, the polarization can be either +1 or -1. The potential energy Vz

comes from a time-varying electric field, 0(t), and the kinetic energy term, in this two-state basis, now has the form KF ,x

where F is the Pauli matrix. Since F is off-diagonal in the polarization basis, this term contains the (nonlinear) couplingx x

between the states of the quantum dot molecule at successive time slices. The size of this term, given by the parameter K(the “tunneling amplitude”), is determined by the physics of the dot molecule: how easy it is for the electrons to tunnel frompolarization state +1 to -1. The effect of the optical phonons is summarized by the influence functional I[F (t)], given byz

I[Fz(t)] ' m kk D["k(t)] exp( iSm

T

0

dJ jk

[mk

20"2

k(J)%mkT2

k

2"2

k(J)%8k"k(J)Fz(J) ] )

I [Fz(t)] ' exp[jjjj )Fz(j)t)P(j)t,j )t)Fz(j )t)]

P(J )'jk

82k

2mkTk

csch($STk/2)[cosh($STk/2)cos(TkJ )%isinh($STk/2)sin(TkJ )]

Output'Th(|<%|R(Fz(N)t),T)>|2)

�'14

[|<%|R>|2&Desired]2

3

(4)

(5)

(6)

(7)

(8)

where " is the position variable of the k harmonic oscillator (phonon), m its mass, T its frequency, and 8 its couplingk k k kth

strength to the system. The advantage of a linearly coupled harmonic bath is that the path integrals over the phonons canbe performed immediately, giving us the nonlinear functional:

where P(J, J’) = P(|J-J’ |) = P(J’‘) is the influence phase, proportional to the response function of the bath. For the phononbath,

where we have introduced also a (suitably low) temperature, given by 1/$ in units of Boltzmann’s constant. Since I[F (t)]z

is a functional of the state of the quantum system at all times (i.e. the activation levels of all the neurons), it adds anothernonlinearity. The trainable parameters can be taken to be the coupling strengths to each of the bath oscillators, {8 }, thek

values of the electric field at each time slice j, {0 }, the frequencies of the oscillators, {T }, or any combination of these.j k

Each of these can be controlled physically: {8 } , by optically exciting multiple phonons; {, }, by changing the external field,k j

and {T } , by changing the phonon frequencies excited. It should be noted that this is not a feed-forward network: all neuronsk

are connected to all other neurons, both forward and backward in time, by the effects of the environment.

IV. RESULTS AND DISCUSSION

We now set up a simulation of the quantum neural network to do a logic gate. We specify as inputs the initial (t=0)polarizations of each of two quantum dot molecules, far enough from each other spatially that they do not interact directly,but sharing the same substrate. The inputs for the logic gates correspond to specification that the polarizations P of the twomolecules: (0,0) : { -1, -1} ; (0,1) : { -1, +1}; (1,0) : {+1, -1}; and (1,1) : {+1, +1}. So for example, if the two moleculespictured in Figure 1 were our two molecules in their prepared input states, this would correspond to a logical input of (1,0).For output we (arbitrarily) take the polarization of the first molecule at the final time, which could be determined bymeasuring the electric field of the quantum dot molecule, and threshold it at some value. Thus, the computed output of thesystem, which is between 0 and 1, is thresholded:

where <+|R(F (N)t),T) > is the computed probability amplitude for the first molecule’s final state to be equal to the <+|z

state. We consider the network trained if the absolute magnitude squared of the probability amplitude for the first molecule’spolarization is greater than (less than) that chosen threshold value, as desired for the given logic gate. For example, for theXOR the “goals” for the computed outputs from the given inputs would be 0.01, 0.1, 0.1, and 0.01, respectively.

A simple gradient descent algorithm was used to train the network. The error function is given by

where the trace over the state of the second molecule is understood. We then differentiate the Lagrangian with respect to

M�M8k

'12

[|<%|R>|2&Desired][2Re[ M<%|R>M8k

<%|R>(]

8newk '8old

k &0 M�M8k

4

(9)

(10)

Figure 2 quantum neural network. Two quantum dot molecules are evolving in time; each of the pictures as readfrom left to right is a snapshot of the states of the molecules at a given time. The far left states represent the input layer. The states on the far right represent the output layer. In this particular example the inputs are {-1,+1}, corresponding toa logical (0,1). The particular path shown is one of 2 possible paths; here, N=4. This particular path’s probabilityN+1

amplitude would contribute to the <+|R>; the net would be training to increase this contribution if, for example, we aredoing an XOR.

the particular parameter we wish to train; for example, with respect to the coupling strength 8 of the quantum system tok

phonon k we have:

For each training pair, all paths (i.e. all possible states of the set of neurons; the sum over the different values for the set of{ F } in Eq.(3) ) are evaluated exactly, and all the contributions summed, to compute both the result, |<+|R>|, and thez

derivatives of � with respect to each of the adjustable weights. This identifies that direction in the parameter space ofweights which minimizes the error. The weights are then changed in that direction, according to

where 0 is the training rate, and the process repeats. Figure 2 shows a picture of one particular path of the 2 paths, whoseN

amplitude would contribute. The two rows of pictures should be read from left to right as being a series of snapshots of whatthe state of each of the two molecules look like, at each time slice. Each time slice corresponds to a neuron. The far leftstates, in this case {-1,1}, correspond to the input layer; the far right states, to the output layer.

Figure 3 shows the error as a function of training pass, for one logic gate, the XOR, and Figure 4 shows theasymptotic error value as a function of the number of neurons, N. For N < 5 the XOR gate does not train satisfactorily, butfor N=5 or 6 the error is essentially zero. By adjusting the values of the tunneling amplitude K, the training rates 0 for theelectric field and for the coupling strengths, the thresholding values, and the characteristic frequencies T , we have beenk

able to find regions in which we can train the net successfully to do any classical logic gate. Results are shown in Table 1for one of these regions, with discretization n equal to 5 points (neurons.) Each of these calculations was started at the samepoint in the parameter space, as indicated in the table.

Because the time dynamics are quantum mechanical in nature, the network is also capable of performing purelyquantum computation, such as phase shifts, which cannot be done by a classical computer. Preliminary calculations, done

5

Figure 3 Error as a function of training pass, for the XOR gate. A three-oscillator bath was used, with frequencies of1.3, 1.06, and 1.65 (in units of 1.5 x 10s). The temperature was 1.2 mK, the tunneling amplitude between states 610

meV, the total time of evolution 1 psec. The number of discretization points (neurons) was 5. The training rate for thefields was 5 x 10, and for the couplings 1 x 10. -4 -5

using a single quantum dot molecule (one input, one output), show that the net is indeed capable of these; however, for thisto be implemented physically we would need a sufficiently sensitive method for measuring the phase [9]. The net is alsocapable of doing computation forwards or backwards in time: computationally, this means simply replacing time withnegative time. Since in this model no dissipative influence was included, the net is symmetric in time; thus, in principle atleast, the net could provide the answer before the question was asked.

V. CONCLUSIONS AND FUTURE WORK

Potentially, a quantum neural network would be an extremely powerful computational tool. Moreover it is capable,at least in principle, of performing computations that cannot be done, classically. This current work, of course, merelydemonstrates proof of concept; an actual working quantum neural net would likely want to take advantage of the greatermultiplicity and connectivity inherent in an entire array of quantum dot molecules, by placing the molecules physically closeenough to each other that nearest neighbors can interact directly, as Lent, Tougaw and Porod [4] proposed for theiralgorithmic quantum dot computer. This would have the additional advantage of reducing error, in the sense that if onemolecule becomes damaged, the net can correct for the resulting error; this mechanism is not possible with the presentarchitecture.

6

Figure 4 Assymptotic error as a function of N, the number of neurons, for the XOR gate. Same parameters andprocedures as for Figure 3.

The major question that still needs to be addressed, is: what level of noise is sufficient to destroy the quantumcoherence and thus the computational power of the quantum neural net? Roughly, this will occur when the dissipation timescale approaches the computational time. This work is currently in progress.

VI. ACKNOWLEDGEMENT

This work was supported in part by the National Science Foundation, under grant ECS-9312345, and by the University ofKansas Center for Research, Inc.

VII. REFERENCES

1. P.D. Wasserman, in Neural Computing Theory and Practice. Van Nostrand Reinhold, New York, 1965.

2. S.R. Skinner, E.C. Behrman, A.A. Cruz-Cabrera, and J.E. Steck, "Neural network implementation using self-lensing media," Appl. Opt., vol 34, pp. 4129-4135, 10 Jul. 1995; R.R. Birge, "Protein-based computers," Sci.Am., vol272, pp. 90-95, Mar. 1995.

3. A.Barenco, D. Deutsch, A. Ekert, and R. Jozsa, "Conditional quantum dynamics and logic gates," Phys. Rev. Lett.,vol 74, pp. 4083-4086, 15 May 1995; J.J. Cirac and P. Zoller, "Quantum computations with cold trapped ions,"Phys. Rev. Lett., vol 74, pp. 4091-4094, 15 May 1995.

4. C.S. Lent, P.D. Tougaw, and W. Porod,"Quantum cellular automata: the physics of computing with arrays ofquantum dot molecules," in Proceedings of the Workshop on Physics and Computation (PhysComp '94), IEEE

7

Computer Society Press, 1994, pp. 5-13.

5. R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals. McGraw-Hill, New York, 1965.

6. M. Kemerink and L.W. Molenkamp,"Stochastic Coulomb blockade in a double quantum dot," Appl. Phys. Lett.,vol. 65, pp. 10 12- 1014, 22 Aug. 1994.

7. A. O. Caldeira and A.J. Leggett, "Quantum tunneling in a dissipative system", Ann. Phys., vol 149, pp.374-456,1983.

8. Y. Wan, G. Ortiz, and P. Phillips, "Pair tunneling in semiconductor quantum dots," Phys. Rev. Lett., vol 75,pp.2879-2882, 9 Oct. 1995.

9. J.H. Shapiro, S.R. Shepard, and N.C. Wong, "Ultimate quantum limits on phase measurement," Phys. Rev. Lett.,vol. 62, pp. 2377-2380, 15 May 1989.

Table 1: Quantum neural network binary classical logic gates. Conditions were the same as in Figure 3.

INPUT 0 0 0 1 1 0 1 1GATE

Goal 0.01 0.10 0.10 0.10OR Start 0.2032 0.0426 0.3331 0.0964

End 0.0110 0.1338 0.3486 0.1816

Goal 0.10 0.01 0.01 0.01NOR Start 0.2032 0.0426 0.3331 0.0964

End 0.3287 0.0108 0.0110 0.0110

Goal 0.01 0.01 0.01 0.10AND Start 0.2032 0.0426 0.3331 0.0964

End 0.0114 0.0129 0.0139 0.6499

Goal 0.10 0.10 0.10 0.01NAND Start 0.2032 0.0426 0.3331 0.0964

End 0.2088 0.1003 0.2150 0.0110

Goal 0.01 0.10 0.10 0.01XOR Start 0.2032 0.0426 0.3331 0.0964

End 0.0110 0.1007 0.3358 0.0110

Goal 0.10 0.01 0.01 0.10XNOR Start 0.2032 0.0426 0.3331 0.0964

End 0.1030 0.0102 0.0158 0.1023

Goal 0.10 0.01 0.10 0.01SPEC Start 0.2032 0.0426 0.3331 0.0964

End 0.0539 0.0093 0.2350 0.0103

Goal 0.01 0.10 0.01 0.10NSPEC Start 0.2032 0.0426 0.3331 0.0964

End 0.0089 0.2233 0.0103 0.2735