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Quantum noise in reversible soliton logic
Optics Communications North-Holland
OPTICS COMMUNICATIONS
105 (1994) 99-103
Quantum noise in reversible soliton logic P.D. Drummond
and W. Man
Department of Physics, The University of Queensland, Queensland 4072, Australia Received 25 August 1993
Reversible logic gates using soliton collisions were recently demonstrated. These are of interest as an example of non-dissipative switching. We show that these gates are unusually susceptible to quantum noise or vacuum fluctuation effects. In fact, with typical experimental parameters we predict error rates approaching 50°h after just 200 operations, __ entirely due to intrinsic quantum noise.
The idea of reversible logic gates has been used to demonstrate the lack of a fundamental energy requirement in performing logical operations [ 11. In this Letter, we show that synchronous logic can develop a more subtle problem, due to a sensitivity to quantum noise. This means that quantum effects can give rise to large error rates after relatively few logical operations, even for macroscopic signal levels. In order to illustrate this problem, it is essential to use a realistic quantum model, although idealized model quantum computers [ 21 have been treated previously. We choose for this purpose, a recently demonstrated type of reversible logic [ 31, involving collisions of fiberoptic solitons. We find large error-rates due to intrinsic quantum noise effects. Quantum effects in coherent fiber-optic solitons are known to cause quantum phase-diffusion and squeezing [ 4-71. This effect was also calculated using linearized operator methods [8] and has been experimentally verified [ 91. Position diffusion [8, lo] is also known to occur in theory. A recent review of quantum solitons demonstrates that these effects are generic in coherent soliton propagation, even without amplification [ 111. No experimental results for position diffusion are available as yet for non-amplified signals. The effect of soliton collisions has been treated theoretically [ 12,131, and observed in experiment as a type of QND measurement [ 141. We now wish to use these techniques to analyse solitonic pulse-position logic. In the limit of high photon number, fiber-optical solitons can be treated using the
semi-classical Wigner equation [ 15 ] which approximates the quantum nonlinear Schrodinger equation. The Wigner equation has the same dynamical form as a classical nonlinear Schrodinger equation,
(1) Here T = vt/xo and X = (x/v - t)/to are the scaled time and position, respectively, in a frame moving at the group velocity v = v (00 ), relative to some reference frequency 00. The parameter to is a typical pulse duration, while xc = ti/ Ik”( is the characteristic interaction length defined by the (anomalous) dispersion parameter k” = d2k ~~w*~~=~~. The dimensionless photon density in these scaled units is Z (Ul’, where 7i = lk”l AC/ (f2nzw&) is a characteristic photon number scale. As usual, A is the effective modal cross-sectional area, while nz is the nonlinear refractive index in units of (m’/W). In laser experiments involving colliding solitons, we can assume the initial quantum state is a coherent state. Quantum effects in this state come from the initial vacuum fluctuations in the Wigner representation [ 161. The initial fluctuations are therefore correlated according to (AU(O,X)AU(O,X’)) (AU(O,X)AU*(O,X’))
= 6(X - X’)/(2A).
From the information above, the one-soliton tions for eq. ( 1) are given at T = 0 by
0030-4018/94/S 07.00 0 1994 Elsevier Science B.V. All rights reserved. SSDI 0030-4018(93)EOSSS-T
= 0, (2) solu-
99
U=u+AU.
On linearizing
(3)
Here ?? is a classical soliton, evolve in time as u = ;?sech[8(T,X)]
that would classically
exp[i?!(T,X)],
is
+ iIf%
e(r,x)
= A(X-X-VT).
+ is
+ i(?
[ 4,5,17]
+ (21V1* - iA*)U = 0.
This can be transformed au iar
=7(X-X-77-)
eq. ( 1)) we obtain
+ U*u*exp(-2ia)
(4)
with 8, E defined as E(T,X)
15 January 1994
OPTICS COMMUNICATIONS
Volume 105, number 1,2
(7)
into the form
+ A*Lu = 0,
(8)
+;i2)z-+F, where the vector u and matrix operator L are defined (5)
In this equation we have the usual interpretation that 1 is a classical amplitude, 7 is a classical velocity, 7 is an initial position, and ij5 is an initial phase. In order to solve these equations perturbatively, we now consider the effect of small fluctuations in the initial conditions. This procedure is valid for quantum noise if Ti >> 1 and the delta-function in eq. (2) is regarded as a tempered function with a momentum cut-off, so that the perturbation is in fact small. A comparison of the cut-off calculation in the Wigner representation, with a more precise calculation without divergent noise (in the positive P-representation) is given elsewhere [ 16 1. Following I&up’s [ 17 ] perturbation theory method, we define an expansion parameter E = ?I-‘/*, and allow the solutions to be expanded as U = V + euexp[icu(T,X)],
by u=
u [ u* I ’
L = ju3 (a,’ - 1) + (21~s + ia2) sech*(8).
Using standard notation, gt, r~2,03are Pauli spin matrices. The solution u can be expanded in a complete set of eigenstates of the operator L, u(T,X)
dk
= s
x ]s(k,T)u(k,0)
+
h(k,T)o,u(kO)l
+ ~gj(T)u’j’(O).
(10)
Here u(k), u(‘~*) are the eigenstates of L, while ~(~3~) are derivative states included to allow closure. The states u(j) satisfy
(6)
where
LIP’ = 0, Lu'4'
=
Lu’2’ = 0,
L/4(3’ = u(l),
_u(*),
(11)
6 = Asech[8(T,X)]exp[icu(T,X)]. Here the parameters A, I’, I, F used in the functions 8, (Yare chosen to remove secular terms from the resulting perturbation theory for u, giving new soliton parameters that include part of the initial quantum noise AU (0, X). The remaining quantum noise is included in the term u, which behaves as a continuum term. We will show that the new parameters are only sensitive to low-frequency components of the initial condition noise, to first order in perturbation theory. This means that they are insensitive to any momentum cut-off, as expected. Higher order perturbation theory requires a renormalisation in the Wigner representation, since there are cut-off dependent terms of order (6’) in the Wigner equations - which we have neglected here. 100
(9)
and are given explicitly Qup U(I) =
$
$ech(e)
u(3) =
=
1 -1
sech(e)
U(2) = d4)
in a notation
]171, by
[
I
tanh(B)
sech(e)
,
1
[ 1.
i
= 1)
,
[I
1 -1
The states that have nonvanishing (u(n)l*(n**))
from
, 1 [I
(l-8tanh8)sech(C?)
e
adapted
(12) inner products are (13)
Volume 105, number 1,2
This results in the following non-vanishing tions in the initial parameters,
where (I+)
15 January 1994
OPTICS COMMUNICATIONS
E 7
df3ur(0)
asv(6). (AA’) = A/(2A),
-cc Next, we note that to the same (linear) order in E, the initial fluctuations can be equated with the change in parameters through a first order Taylor expansion in E, carried out at T = 0,
+ ~u(O,X)exp[ia(O,X)l,
(14)
where AA = (A - 2) = (AA, AV, A$, AZ). Here we have assumed that AA = 0 (E ) . The derivative terms are obtained to be VA. 6 = (~,‘~‘,i~i~‘,2iAul”,2A[Aul”
- iVui”])
x exp(icu).
(15)
We now take inner products
of
(16) with the four discrete eigenvectors u(j), to obtain the value of AAi, noting that the condition of no secular terms implies that gj (7’) E 0. This is necessary to ensure that the term u (T, X) has no components that increase linearly with time. Using the orthogonality and inner product properties of the states u(j), we obtain AA = Re
AUexp(-icu)
,
seched0 >
(I AV = Re
-iAU exp( -ia)
,
sech6 tanh 8 de
AU exp( -icr) (I
x [(i/A)(etanhf3-
,
1) + (V/A2)8]sech0dt9 >
A_? = Re
AU exp( -ia) (J
8 seche de
A2.
(AV2) = A/(6@,
(A$2) = (1 + a2/12 + V2n2/4A2)/(6EA), (AT2) = n2/(24EA3), (A$Ax)
= Vn2/(24EA3).
(18)
In summary, neglecting the continuum terms, a coherent input soliton behaves as though quantum noise effects cause gaussian fluctuations in all the initial soliton parameters. We note that this results in time-dependent behaviour in measurements taken at times long after the time when the soliton is injected into the fiber at T = 0. Thus, in agreement with exact quantum results and some earlier calculations using operator perturbation theory, we expect that the mean soliton position and phase variance should increase in time. This follows since d(T) = 4(O) + $(A2 + V2)T and x(T)
= x(O)
(ti2(T))
= (a2/4A2
(Ad2 (T))
= (1 + 7r2/12 + V2n2/2A2
+ VT, hence + A2T2)/
(6EA) ,
+ V2A2T2 + 3A4T2)/(6TiA).
(19)
There is also a correlation between Ax and A4 at finite velocity V, owing to the presence of terms proportional to VA2 in the phase of the perturbed soliton. If two solitons are well separated, the addition of two one-soliton solutions will be a good approximation for the two-soliton solution [ 181. The initial expression for the two soliton solutions is therefore given approximately by u = u, +
>
(J AI#J= Re
correla-
u2,
(20)
where 17, and UZ are one-soliton solutions for each independent soliton. The position shift of soliton ( 1) after a complete collision with soliton (2 ) is given by the following collision formula [ 18,191, SF, = WY, + (l/A,
) ln(D-/D+
D* = (K-q)‘+
(A2fA1)‘.
1,
>/ (17)
(21) 101
Volume 105, number
15 January 1994
OPTICS COMMUNICATIONS
1,2
(24) Noting that A, V are invariant under collisions, overall position shift resulting is AXi
Fig. 1. Numerically generated graph of pulse amplitude versus X and T for a typical collision, using data given in the text. Unit width input solitons have locations X = 0, X = -8, and velocities of V = 0, V = 0.1. Total propagation distance is T= 60. This gives results that agree well with the collision formula, eq. (21).
Here At, Az and Vi, VZare the amplitudes and velocities of soliton ( 1) and (2)) while W: is the position shift due to the collision between soliton ( 1) and the continuous spectrum perturbed from soliton (2). A numerically calculated graph of a typical collision, where initially Vz > Vi and X2 < Xi, is given in fig. 1. The continuous spectrum term is WY1 = -G
1
co s
ln(1 + Ir(k)12) dk, (k - Vl,/2)2 + A?/4
(22)
where r(k) is a reflection coeffkient in the inverse scattering formalism, of order E. In a calculation to order E, the term Wzl is clearly
= AX,(O) + AV, T + A&?r.
the (25)
Here we neglect the noise introduced by the collision between soliton ( 1) and the continuous spectrum of soliton (2). This is small compared with all the other terms, because the contribution due to the quantum noise in the continuous spectrum is proportional to the square of the reflection coefficient [ 18,191, i.e., approximately proportional to e2. With soliton (1) having zero average velocity in the frame of reference (7, = 0), the total variance of soliton position after one collision is given to the lowest order by
(26) Each logic operation will involve either one or zero collisions between the soliton pulses. On the average, approximately half the number of logic operations will involve collisions. Therefore, the typical position fluctuation A&, of a particular logic soliton after a sequence of N pulse position logic operations can be given by the following expression, AX,,(TN) = AX,,(O) + AV, T
negligible. With the result given above, the central position fluctuation in soliton ( 1) due to the collision may be expanded to order t as follows (27)
The soliton position given by
variance after N operations
is
(23) (AXON)
where ---
1 DA; D+ ’
=
&
+$
+g[(gy&+(gp] N
dS~oAT
+iFlm
.
(28)
Volume 105, number 1,2
OPTICS COMMUNICATIONS
It is clear from this equation that as long as each logic operation takes a time duration T >> 1, the overall noise is dominated by the first term. This is due to wave-packet spreading, or equivalently to initial frequency shot-noise in the logic pulse. Noise due to collisions is relatively small, except for the last term. This is caused by a cross-correlation between the velocity and collisional position-shift, and tends to reduce overall fluctuations slightly. Despite this, the overall trend is a quantum wave-packet spreading that gives a variance increasing quadratically with elapsed time. This type of logic therefore gives rapidly increasing error-rates as the number of logic operations increase, since reversible logic provides no mechanism for resynchronizing the pulse positions. We conservatively assume that the logic will fail if the soliton position fluctuation increases to the half of the pulse width, so we set the left hand side of eq. (28) equal $. Using typical experimental data of A, = 1, Vn - I’ = 0.1, and X = 6 pulse widths for each complete operation, and ii = lo*, we find the time taken for each logic operation is T = 60. The number of logic operations that can be performed before the logic fails can be obtained by solving the following quadratic equation, ii n2 -=24 + 600N2 + 33.6N. 4
(29)
Using Ii = lo’, we find a result of N z 200 operations, which is a relatively small number compared with the number of operations used by a computer programme. Furthermore, the equations describe a perfect lossless libre without any other source of noise, so that the performance of the logic gates would be even worse in practice. We conclude that using purely conservative pulse position logic in a real computer network is unlikely to be successful, unless these gates could be combined with other techniques that allow a dissipative re-synchronization of signals, before errors accumulate. Similar, or even stronger reservations have been expressed by earlier authors [ 201. However, the use of pulses with sub-shot-noise velocity error (squeezed frequency pulses) could improve these estimates. Other possibilities include Feynmann’s proposal [ 2 1 ] for asynchronous quantum computers, in which the logical result heralds its own arrival - so the quantum uncertainty goes into the arrival time, rather than the result itself.
15 January 1994
Finally, we note that our general analysis is also applicable to QND measurements of phase and position, as well as to reversible phase-logic gates. P.D.D. would like to acknowledge a number of useful discussions with Y. Yamamoto, J. Breslin, R. M. Shelby and S. R. Friberg.
References [I] C.H. Bennet, IBM J. Res. Dev. 17 (1973) 525; E. Fredkin and T. Toffoli, Intern. J. Theor. Phys. 21 (1982) 219; C.H. Bennet, IBM J. Res. Dev. 32 (1988) 16. [2] P. Bennioff, J. Stat. Phys. 22 (1980) 563; 29 (1982) 515; Phys. Rev. Lett. 48 (1982) 1581. [ 31 M.N. Islam and C.E. Soccolich, Optics Lett. 16 (1991) 1490. [4] S.J. Carter, P.D. Drummond, M.D. Reid and R.M. Shelby, Phys. Rev. Lett. 58 (1987) 1841. [ 51 P.D. Drummond and S.J. Carter, J. Opt. Sot. Am. B 4 (1987) 1565. [6] P.D. Drummond and S.J. Carter, Optics Lett. 14 (1987) 373. [ 71 R.M. Shelby, P.D. Drummond and S.J. Carter, Phys. Rev. A 42 (1990) 2966. [8] H.A. Haus and Y. Lai, J. Opt. Sot. Am. B 7 (1990) 386. [9] M. Rosenbluth and R.M. Shelby, Phys. Rev. Lett. 66 (1991) 153. [lo] J.P. Gordon and H.A. Haus, Optics Lett. 77 (1986) 665. [ 111 P.D. Drummond, R. M. Shelby, S.R. Friberg and Y. Yamamoto, Nature 365 (1993) 307. [ 121 K. Watanabe, H. Nakano, A. Honold and Y. Yamamoto, Phys. Rev. Lett. 62 (1989) 2257. [ 131 H.A. Haus, K. Watanabe and Y. Yamamoto, J. Opt. Sot. Am. B 6 (1989) 1138. [ 141 S.R. Friberg, S. Machida and Y. Yamamoto, Phys. Rev. Lett. 69 (1992) 3165. [ 151 E.P. Wigner, Phys. Rev. 40 (1932) 749. [ 161 P.D. Drummond and A.D. Hardman, Europhys. Lett. 21 (1993) 270. [ 171 D.J. Kaup, Phys. Rev. A 42 (1990) 5689. [ 181 V.E. Zakharov and A.B. Shabat, Sov. Phys. JETP 34 (1972) 62. [ 191 L. Martinez Alonso, Phys. Rev. D 32 (1985) 1459. [20] R. Landauer, Physica A 168 (1990) 75. [2 1] R. Feynmann, Found. Phys. 16 ( 1986) 507.
103
OPTICS COMMUNICATIONS
105 (1994) 99-103
Quantum noise in reversible soliton logic P.D. Drummond
and W. Man
Department of Physics, The University of Queensland, Queensland 4072, Australia Received 25 August 1993
Reversible logic gates using soliton collisions were recently demonstrated. These are of interest as an example of non-dissipative switching. We show that these gates are unusually susceptible to quantum noise or vacuum fluctuation effects. In fact, with typical experimental parameters we predict error rates approaching 50°h after just 200 operations, __ entirely due to intrinsic quantum noise.
The idea of reversible logic gates has been used to demonstrate the lack of a fundamental energy requirement in performing logical operations [ 11. In this Letter, we show that synchronous logic can develop a more subtle problem, due to a sensitivity to quantum noise. This means that quantum effects can give rise to large error rates after relatively few logical operations, even for macroscopic signal levels. In order to illustrate this problem, it is essential to use a realistic quantum model, although idealized model quantum computers [ 21 have been treated previously. We choose for this purpose, a recently demonstrated type of reversible logic [ 31, involving collisions of fiberoptic solitons. We find large error-rates due to intrinsic quantum noise effects. Quantum effects in coherent fiber-optic solitons are known to cause quantum phase-diffusion and squeezing [ 4-71. This effect was also calculated using linearized operator methods [8] and has been experimentally verified [ 91. Position diffusion [8, lo] is also known to occur in theory. A recent review of quantum solitons demonstrates that these effects are generic in coherent soliton propagation, even without amplification [ 111. No experimental results for position diffusion are available as yet for non-amplified signals. The effect of soliton collisions has been treated theoretically [ 12,131, and observed in experiment as a type of QND measurement [ 141. We now wish to use these techniques to analyse solitonic pulse-position logic. In the limit of high photon number, fiber-optical solitons can be treated using the
semi-classical Wigner equation [ 15 ] which approximates the quantum nonlinear Schrodinger equation. The Wigner equation has the same dynamical form as a classical nonlinear Schrodinger equation,
(1) Here T = vt/xo and X = (x/v - t)/to are the scaled time and position, respectively, in a frame moving at the group velocity v = v (00 ), relative to some reference frequency 00. The parameter to is a typical pulse duration, while xc = ti/ Ik”( is the characteristic interaction length defined by the (anomalous) dispersion parameter k” = d2k ~~w*~~=~~. The dimensionless photon density in these scaled units is Z (Ul’, where 7i = lk”l AC/ (f2nzw&) is a characteristic photon number scale. As usual, A is the effective modal cross-sectional area, while nz is the nonlinear refractive index in units of (m’/W). In laser experiments involving colliding solitons, we can assume the initial quantum state is a coherent state. Quantum effects in this state come from the initial vacuum fluctuations in the Wigner representation [ 161. The initial fluctuations are therefore correlated according to (AU(O,X)AU(O,X’)) (AU(O,X)AU*(O,X’))
= 6(X - X’)/(2A).
From the information above, the one-soliton tions for eq. ( 1) are given at T = 0 by
0030-4018/94/S 07.00 0 1994 Elsevier Science B.V. All rights reserved. SSDI 0030-4018(93)EOSSS-T
= 0, (2) solu-
99
U=u+AU.
On linearizing
(3)
Here ?? is a classical soliton, evolve in time as u = ;?sech[8(T,X)]
that would classically
exp[i?!(T,X)],
is
+ iIf%
e(r,x)
= A(X-X-VT).
+ is
+ i(?
[ 4,5,17]
+ (21V1* - iA*)U = 0.
This can be transformed au iar
=7(X-X-77-)
eq. ( 1)) we obtain
+ U*u*exp(-2ia)
(4)
with 8, E defined as E(T,X)
15 January 1994
OPTICS COMMUNICATIONS
Volume 105, number 1,2
(7)
into the form
+ A*Lu = 0,
(8)
+;i2)z-+F, where the vector u and matrix operator L are defined (5)
In this equation we have the usual interpretation that 1 is a classical amplitude, 7 is a classical velocity, 7 is an initial position, and ij5 is an initial phase. In order to solve these equations perturbatively, we now consider the effect of small fluctuations in the initial conditions. This procedure is valid for quantum noise if Ti >> 1 and the delta-function in eq. (2) is regarded as a tempered function with a momentum cut-off, so that the perturbation is in fact small. A comparison of the cut-off calculation in the Wigner representation, with a more precise calculation without divergent noise (in the positive P-representation) is given elsewhere [ 16 1. Following I&up’s [ 17 ] perturbation theory method, we define an expansion parameter E = ?I-‘/*, and allow the solutions to be expanded as U = V + euexp[icu(T,X)],
by u=
u [ u* I ’
L = ju3 (a,’ - 1) + (21~s + ia2) sech*(8).
Using standard notation, gt, r~2,03are Pauli spin matrices. The solution u can be expanded in a complete set of eigenstates of the operator L, u(T,X)
dk
= s
x ]s(k,T)u(k,0)
+
h(k,T)o,u(kO)l
+ ~gj(T)u’j’(O).
(10)
Here u(k), u(‘~*) are the eigenstates of L, while ~(~3~) are derivative states included to allow closure. The states u(j) satisfy
(6)
where
LIP’ = 0, Lu'4'
=
Lu’2’ = 0,
L/4(3’ = u(l),
_u(*),
(11)
6 = Asech[8(T,X)]exp[icu(T,X)]. Here the parameters A, I’, I, F used in the functions 8, (Yare chosen to remove secular terms from the resulting perturbation theory for u, giving new soliton parameters that include part of the initial quantum noise AU (0, X). The remaining quantum noise is included in the term u, which behaves as a continuum term. We will show that the new parameters are only sensitive to low-frequency components of the initial condition noise, to first order in perturbation theory. This means that they are insensitive to any momentum cut-off, as expected. Higher order perturbation theory requires a renormalisation in the Wigner representation, since there are cut-off dependent terms of order (6’) in the Wigner equations - which we have neglected here. 100
(9)
and are given explicitly Qup U(I) =
$
$ech(e)
u(3) =
=
1 -1
sech(e)
U(2) = d4)
in a notation
]171, by
[
I
tanh(B)
sech(e)
,
1
[ 1.
i
= 1)
,
[I
1 -1
The states that have nonvanishing (u(n)l*(n**))
from
, 1 [I
(l-8tanh8)sech(C?)
e
adapted
(12) inner products are (13)
Volume 105, number 1,2
This results in the following non-vanishing tions in the initial parameters,
where (I+)
15 January 1994
OPTICS COMMUNICATIONS
E 7
df3ur(0)
asv(6). (AA’) = A/(2A),
-cc Next, we note that to the same (linear) order in E, the initial fluctuations can be equated with the change in parameters through a first order Taylor expansion in E, carried out at T = 0,
+ ~u(O,X)exp[ia(O,X)l,
(14)
where AA = (A - 2) = (AA, AV, A$, AZ). Here we have assumed that AA = 0 (E ) . The derivative terms are obtained to be VA. 6 = (~,‘~‘,i~i~‘,2iAul”,2A[Aul”
- iVui”])
x exp(icu).
(15)
We now take inner products
of
(16) with the four discrete eigenvectors u(j), to obtain the value of AAi, noting that the condition of no secular terms implies that gj (7’) E 0. This is necessary to ensure that the term u (T, X) has no components that increase linearly with time. Using the orthogonality and inner product properties of the states u(j), we obtain AA = Re
AUexp(-icu)
,
seched0 >
(I AV = Re
-iAU exp( -ia)
,
sech6 tanh 8 de
AU exp( -icr) (I
x [(i/A)(etanhf3-
,
1) + (V/A2)8]sech0dt9 >
A_? = Re
AU exp( -ia) (J
8 seche de
A2.
(AV2) = A/(6@,
(A$2) = (1 + a2/12 + V2n2/4A2)/(6EA), (AT2) = n2/(24EA3), (A$Ax)
= Vn2/(24EA3).
(18)
In summary, neglecting the continuum terms, a coherent input soliton behaves as though quantum noise effects cause gaussian fluctuations in all the initial soliton parameters. We note that this results in time-dependent behaviour in measurements taken at times long after the time when the soliton is injected into the fiber at T = 0. Thus, in agreement with exact quantum results and some earlier calculations using operator perturbation theory, we expect that the mean soliton position and phase variance should increase in time. This follows since d(T) = 4(O) + $(A2 + V2)T and x(T)
= x(O)
(ti2(T))
= (a2/4A2
(Ad2 (T))
= (1 + 7r2/12 + V2n2/2A2
+ VT, hence + A2T2)/
(6EA) ,
+ V2A2T2 + 3A4T2)/(6TiA).
(19)
There is also a correlation between Ax and A4 at finite velocity V, owing to the presence of terms proportional to VA2 in the phase of the perturbed soliton. If two solitons are well separated, the addition of two one-soliton solutions will be a good approximation for the two-soliton solution [ 181. The initial expression for the two soliton solutions is therefore given approximately by u = u, +
>
(J AI#J= Re
correla-
u2,
(20)
where 17, and UZ are one-soliton solutions for each independent soliton. The position shift of soliton ( 1) after a complete collision with soliton (2 ) is given by the following collision formula [ 18,191, SF, = WY, + (l/A,
) ln(D-/D+
D* = (K-q)‘+
(A2fA1)‘.
1,
>/ (17)
(21) 101
Volume 105, number
15 January 1994
OPTICS COMMUNICATIONS
1,2
(24) Noting that A, V are invariant under collisions, overall position shift resulting is AXi
Fig. 1. Numerically generated graph of pulse amplitude versus X and T for a typical collision, using data given in the text. Unit width input solitons have locations X = 0, X = -8, and velocities of V = 0, V = 0.1. Total propagation distance is T= 60. This gives results that agree well with the collision formula, eq. (21).
Here At, Az and Vi, VZare the amplitudes and velocities of soliton ( 1) and (2)) while W: is the position shift due to the collision between soliton ( 1) and the continuous spectrum perturbed from soliton (2). A numerically calculated graph of a typical collision, where initially Vz > Vi and X2 < Xi, is given in fig. 1. The continuous spectrum term is WY1 = -G
1
co s
ln(1 + Ir(k)12) dk, (k - Vl,/2)2 + A?/4
(22)
where r(k) is a reflection coeffkient in the inverse scattering formalism, of order E. In a calculation to order E, the term Wzl is clearly
= AX,(O) + AV, T + A&?r.
the (25)
Here we neglect the noise introduced by the collision between soliton ( 1) and the continuous spectrum of soliton (2). This is small compared with all the other terms, because the contribution due to the quantum noise in the continuous spectrum is proportional to the square of the reflection coefficient [ 18,191, i.e., approximately proportional to e2. With soliton (1) having zero average velocity in the frame of reference (7, = 0), the total variance of soliton position after one collision is given to the lowest order by
(26) Each logic operation will involve either one or zero collisions between the soliton pulses. On the average, approximately half the number of logic operations will involve collisions. Therefore, the typical position fluctuation A&, of a particular logic soliton after a sequence of N pulse position logic operations can be given by the following expression, AX,,(TN) = AX,,(O) + AV, T
negligible. With the result given above, the central position fluctuation in soliton ( 1) due to the collision may be expanded to order t as follows (27)
The soliton position given by
variance after N operations
is
(23) (AXON)
where ---
1 DA; D+ ’
=
&
+$
+g[(gy&+(gp] N
dS~oAT
+iFlm
.
(28)
Volume 105, number 1,2
OPTICS COMMUNICATIONS
It is clear from this equation that as long as each logic operation takes a time duration T >> 1, the overall noise is dominated by the first term. This is due to wave-packet spreading, or equivalently to initial frequency shot-noise in the logic pulse. Noise due to collisions is relatively small, except for the last term. This is caused by a cross-correlation between the velocity and collisional position-shift, and tends to reduce overall fluctuations slightly. Despite this, the overall trend is a quantum wave-packet spreading that gives a variance increasing quadratically with elapsed time. This type of logic therefore gives rapidly increasing error-rates as the number of logic operations increase, since reversible logic provides no mechanism for resynchronizing the pulse positions. We conservatively assume that the logic will fail if the soliton position fluctuation increases to the half of the pulse width, so we set the left hand side of eq. (28) equal $. Using typical experimental data of A, = 1, Vn - I’ = 0.1, and X = 6 pulse widths for each complete operation, and ii = lo*, we find the time taken for each logic operation is T = 60. The number of logic operations that can be performed before the logic fails can be obtained by solving the following quadratic equation, ii n2 -=24 + 600N2 + 33.6N. 4
(29)
Using Ii = lo’, we find a result of N z 200 operations, which is a relatively small number compared with the number of operations used by a computer programme. Furthermore, the equations describe a perfect lossless libre without any other source of noise, so that the performance of the logic gates would be even worse in practice. We conclude that using purely conservative pulse position logic in a real computer network is unlikely to be successful, unless these gates could be combined with other techniques that allow a dissipative re-synchronization of signals, before errors accumulate. Similar, or even stronger reservations have been expressed by earlier authors [ 201. However, the use of pulses with sub-shot-noise velocity error (squeezed frequency pulses) could improve these estimates. Other possibilities include Feynmann’s proposal [ 2 1 ] for asynchronous quantum computers, in which the logical result heralds its own arrival - so the quantum uncertainty goes into the arrival time, rather than the result itself.
15 January 1994
Finally, we note that our general analysis is also applicable to QND measurements of phase and position, as well as to reversible phase-logic gates. P.D.D. would like to acknowledge a number of useful discussions with Y. Yamamoto, J. Breslin, R. M. Shelby and S. R. Friberg.
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