Debbie Leung (IQC, UW)

w/ Panos Aliferis, Yingkai Ouyang, Man-Hong Yung (IBM) (IQC, UW) (UIUC)

$$: NSERC, CFI, ORF, CIFAR, MITACS, QWorks, ARO, Mike Lazaridis Fellowship(s)

Two measurement problems in Fault-tolerant

Quantum-Computation

Dec 17, 2007. QEC07, USC

Measurement problem 1: * In measurement based quantum computation will 1 error in a measurement outcome cascade to many, overwhelming the ability to perform QEC?

Measurement problem 2: * NonMarkovian noise is modeled as a system-bath Hamiltonian evolution. Its sup-norm should be sub-threshold for FT method to work. Will a sharp measurement, with an obligatorily large amplitude for the system-bath Hamiltonian, spoil the result?

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Plan: interpret the problem away, thus, the problem and result are inevitably trivial

Disclaimer -- this is not on Foundations of QM

Measurement problem 1: * In measurement based quantum computation will 1 error in a measurement outcome cascade to many, overwhelming the ability to perform QECC?

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*Will focus on graph state quantum computation

Graph state Quantum ComputationGSQC: RBB 0301052 Ã 1WQC: RB01Linear optics app: N 0402005

Circuit interpretation CLN04, AL04 w/ gate-teleportation (GC99, ZLC00)

(1) Produce a graph state(2) Apply single measurements

that can be controlled by prior measurement outcomes

Simulate a circuit element-by-elment -- can identify "regions" responsible for each

Graph state simulation of a circuit element O

0/1x1

0/1x2

0/1xn

|+i

|+i

|+i

...

ein

Pein(in)

eout

Peout (O(in))

|+i = |0i + |1i

= controlled-z

Pi = Pauli rotation indexed by i

SO(ein Pein (in)) = (eout Peout O(in))

(will see this is composable)

Composable simulationfor a universal gate set

££

|+i

•

•

Mx •

ab

b©da

XaZb|i

Xb©d Za H|i

d

S(H):

|+i

•

•

Mx •

ab

ab©c

XaZb|i

Xa Zb©c Z|i

c

S(Z):

Z(-1)a

H

a2

b2

Xa1Zb1

Xa2Zb2 |i

S(CP): a1

b1

a1

b1© a2

a2

b2© a1

Xa1Z b1

©a2

Xa2Z b2

©a1 (CP|i)

Goal: perform Z rotation eiZ

H c|0i

|i

Zc|i

Goal: perform Z rotation eiZZ-Telep (ZT)

H c|0i

|i

Zc|i

Goal: perform Z rotation eiZ

c

Zc ei(-1)aZ XaZb |i

XaZb|i H|0i

ei(-1)aZ

Z-Telep (ZT)

Input state = ei(-1)aZ XaZb |i

H c|0i

|i

Zc|i

Goal: perform Z rotation eiZ

c

Zc ei(-1)aZ XaZb |i

XaZb|i H|0i

ei(-1)aZ

= Xa Zc+b eiZ|i

Z-Telep (ZT)

Xa eiZ

= § Xa Zc+b eiZ|i

H c|0i

|i

Zc|i

Goal: perform Z rotation eiZ

c

Zc ei(-1)aZ XaZb |i

XaZb|i H|0i

ei(-1)aZ

= Xa Zc+b eiZ|i

Z-Telep (ZT)

Xa eiZ

= § Xa Zc+b eiZ|i

|+i

•

•

Mx •

ab

ab©c

XaZb|i

Xa Zb©c Z|i

c

S(Z):

Z(-1)a

H

Composable simulationfor state prep & meas

S(|0i):Mz

a

0XaZ0|0i

a0

0S(|+i):Mx

b

X0Zb|+i

0b

Mz

ab

a©cXaZb|i c

S( Mz ):

Mx

ab

a©cXaZb|i c

S( Mx ):

|+i

given

General quantum circuit C

|0i

|0i

|0i

U1

U2

U3

U4

U5

0/1

0/1

0/1

y1

y2

y3

Composing simulations simulates the circuit

S|0i

S|0i

S|0i

0/1

0/1

0/1

y1

y2

y3

SOk ... SO2

SO1 (Pinit(|0ih0|))

= Pfinal Ok ... O2 O1 (|0ih0|)

S

SU1

SU2

SU3

SU4

SU5 S

S

SO(ein Pein (in)) = (eout Peout O(in))

FT Qn 1: Can we achieve FT simply by simulating a FT circuit ? Physical noise ! effective noise in simulated circuit ?

FT Qn 2: Will an error propagate via feedforward ? Say, 1 error corrupts a meas outcome, which may induce

further errors as it is used to condition subsequent op ... ?

AL05: offer an interpretation in which measurement outcomes are error free ...

yes to FT R03, ND04 -- with much work in designing schemes and

full-blown nonMarkovian treatment of meas outcome errors

S(Xa’ Zb’) (or S(I)):

ab

a©a’b©b’

XaZb|i

a’

b’

Xa©a’ Zb©b’ |i

Composable simulationfor Pauli’s

Note: Known Pauli operations : shifts in the classical parts.Unknown Pauli errors : errors in the classical parts.

0. Add each |+i to the graph state only slightly before it’s being measured

~1. Model noisy elementary operations O

storage, gate, or state prep meas

env

UFO

~O »

env

UFO

~O »

where UF = I A0 + i Pi Ai on sys env

Pi = nontrivial Pauli’s

Now working out how physical noise transformed to dealing the noise

: where UFs act

2. Noisy simulation: interperse UFs between ideal operations e.g.

|+i

•

•

Mx •

ab

ab©c

XaZb|i

Xa Zb©c Z|i

c

S(Z):

Z(-1)a

H

• Sub UF = Isys A0 env + i Pi sys Ai env in above, expand

• For @ summand, commute Pi's towards end of simulation

- “Joint I term” : ideal simulation - Else: (1) classical part may flip (2) quantum part may suffer unknown Pauli error

But they’re equivalent !! So classical part is always correct & faults do not propagate but are localized

How physical errors affect each simulation?

Thus, in GSQC:

No error in classical part (they're unknown Pauli errors)

Each elementary faults is also localized within a simulation

Faults of all elementary operations in 1 simulation give a combined fault in that simulated op, and no where else, with effective noise strength · max # locations in a sim £ max strength per elementary op

All FT results for circuits thus apply (including the code design, threshold analysis, and size+depth bounds)

In fact, the composable graph state simulation is like a lowest level encoding in FTQC

Qn: can GS/other teleportation based implementation decohere different fault paths? Need to understand meas better.

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*

Measurement problem 2: * NonMarkovian noise is modeled as a system-bath Hamiltonian evolution. Its sup-norm should be sub-threshold for FT method to work. Will a sharp measurement, with an obligatorily large amplitude for the system-bath Hamiltonian, spoil the result?

Remember fast measurement & classical components should help FT but how to model these?

Modelling sharp measurements:

Q

C

B

M

|xiQ ! |xiQ |xiC |xiM

need to give a copy of the measurement outcome x to the environment to ensure its classicality

g

g

H = HQ + HC + HM + HB + HQC+ HCM + HBM+ HQM + HQB + HBC

|| HQC ||1 = || HCM ||1 = gsignifies how fast the meas is

|| HBM ||1 = quantities how fast adversarial noise can use the meas outcome

|| HQB ||1 = || HQM ||1 = the usual QC noise amplitude

|| HCB ||1 = ¿ Classical Comp noise amplitude

g large if meas fast

Modelling sharp measurements:

Q

C

B

M

g

g

H = HQ + HC + HM + HB + HQC+ HCM + HBM+ HQM + HQB + HBC

|| HQC ||1 = || HCM ||1 = gsignifies how fast the meas is

|| HBM ||1 = quantities how fast adversarial noise can use the meas outcome

|| HQB ||1 = || HQM ||1 = the usual QC noise amplitude

|| HCB ||1 = ¿ Classical Comp noise amplitude

|| e-iHt - V ||1 · t ( + 2 + )

Let t be the meas time(tg ¼ 1), V be ideal meas

Let tO be the time for a gate, noise strength ¼ tO (tOÀt)

Putting in the separation of scale, as long as · tO/tthe threshold is unspoiled.

Qns: do we need to include in the noise amplitude for gate alone?What to expect for in various systems? Decohering fault paths by (un)intentional "random" Pauli's?

esp in FT designs heavily relying on gate teleportation ...