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