Quantum Information Processing: Algorithms, Technologies and
Quantum information processing with prethreshold ... · Quantum information processing with...
Transcript of Quantum information processing with prethreshold ... · Quantum information processing with...
Quantum information processing with prethreshold superconducting qubits�
M. R. Geller, A. T. Sornborger, and P. C. Stancil (UGA) �
J. M. Martinis (UCSB) �C. Benjamin (UGA) , A. Galiautdinov (UCR), �
and E. J. Pritchett (IQC Waterloo)��
work supported by the NSF EAGER & CDI �
www.physast.uga.edu/~mgeller/group.htm
SES computing n qubits
full Hilbert space single-excitation subspace (SES)
! "U!
2n variablesexponential storage
capacity
but 22n gatesnatural Hamiltonians
are one- and two-bodyefficient algorithms poly(n)
number of gates is largeerror correction required
this is the bottleneck forsuperconducting architectures
! "U!
n variables 1 shot!full controllability
of projected H
requiresfully connected qubits ~n2 JJs( )can only run for time !
more qubits required,but this is OK
n = 3 000 , 001 , 010 , 011 , 100 , 101 , 110 , 111{ }
111
011 , 101 , 110
001 , 010 , 100
000
!
single-excitation subspace
SES computing n qubits
full Hilbert space single-excitation subspace (SES)
! "U!
2n variablesexponential storage
capacity
but 22n gatesnatural Hamiltonians
are one- and two-bodyefficient algorithms poly(n)
applications:1. factoring n ! 2100s (not possible)2. general purpose simulationpolynomial speedup for class of simulations
! "U!
n variables 1 shot!full controllability
of projected H
quantum simulation
analog�digital �
(gate-based)�
Aspuru et al, 2005 Greiner et al, 2002
Buluta & Nori, 2009
Feynman, 1982 Lloyd, 1996
quantum simulation
analog�digital �
(gate-based)�
general-purpose�simulator (UGA/UCSB)�
• tied to UCSB phase qubit architecture�• simulates arbitrary time-dependent H’s�• uses and interfaces with conventional QC�
Aspuru et al, 2005 Greiner et al, 2002
Buluta & Nori, 2009
Feynman, 1982 Lloyd, 1996
mapping
H =p1
2
2m1
+P1
2
2M1
!Z1e
2
r1 ! R1+
p2
2
2m2
+P22
2M 2
!Z2e
2
r2 ! R2+Z1Z2e
2
R1 ! R2+
e2
r1 ! r2!
Z2e2
r1 ! R2!
Z1e2
r2 ! R1
Hucsb = !ici†
i=1
n
" ci +12
giji, j=1i# j
n
" K$%& i$ '&
j
%
?�
what we solve given a d-dimensional Hilbert space�
m{ }m=1
d , m m ' = !mm '
and time-dependent Hamiltonian �
Hmm ' (t), t !(ti ,tf ) (real?)�
U ! T e"(i /!) H (t ) dt
ti
tf
#
= lim$t%0
e"(i /!)H (tf )$t & ' ' ' & e"(i /!)H (ti +$t )$te"(i /!)H (ti )$t
time-evolution operator�
! (tf ) =U ! (ti ) we compute U �
n-qubit simulator UCSB phase qubits with tunable inductive coupling �
Hucsb = !i (t)ci†
i=1
n
" ci +12
giji, j=1i# j
n
" (t)K$%& i$ '&
j
% g! !( )
n(n !1)2
n
n(n +1)2
controls
dim(Hucsb ) = 2n not controllable
R. C. Bialczak et al, arXiv:1007.2209
single-excitation subspace
m) ! cm
† 00 " " "0 = 00 " " "1m " " "0 this has dimension n �(number of qubits)�
complete control over real Hamiltonians�in this subspace�
energy/time rescaling
U = e!(i /!)H t = e!(i /!) H
"#$%
&'( ("t ) = e!(i /!)Hqc tqc
coherence limits length �of process simulated�
Hqc !H", tqc ! "t
! GHz
we will have to emulate Hqc�
molecular collisions! ! 106
(can also subtract time-dependent c-number) �
continuous rescaling
t
tqc
!
continuous rescaling
t
tqc
! ! "dtqcdt
> 0
smaller ! is better
tqc (t) is nonlinear
U = Te
!(i /!) H (t ) dtti
t f
"= Te
!(i /!) Hqc (tqc ) dtqctqc ( ti )
tqc ( t f )
"Hqc (tqc ) #
H (t)$(t) t= t (tqc )
exact symmetry of U �
main error: leakage
111
011 , 101 , 110
001 , 010 , 100
000
!
leakage prob ! g
!"#$
%&'2
( 10)4 (UCSB)
3-channel molecular collision uses classically computed potential energy surface �
Na(3s) + He! Na(m) + He (m = 1,2,3)
-40 -20 0 20 40t
0
0.2
0.4
0.6
0.8
1
P1,1
P1,2
P1,3
-40 -20 0 20 40t
0
0.2
0.4
0.6
0.8
1
-40 -20 0 20 40t
0
0.2
0.4
0.6
0.8
1
-40 -20 0 20 40t
0
0.2
0.4
0.6
0.8
1
b=1.0
b=2.2 b=5.5
b=0.6
scaling
classical (single fast processor)time slice t = 2d 3 ns (large d)tcl = 100 ! 2d 3 nsunfavorable dimension dependence(unrelated to exponentially large d problem)d>1000 rare
quantum simulatoreach 100ns run t = 1µstqu = 100 !1µs = 105 nsindependent of d!
Hilbert space dimension d
scaling (continued)
0 5 10 15 20 250
0.5
1
1.5
2
d
t (m
s)
100 qubit machine is 1000 times faster1000 qubit machine is 106times faster!
a practical QC?
tcl
tqu = 0.1ms
E. J. Pritchett et al., arXiv:1008.0701