roll up our sleeves & prove a few things

Applications of randomized techniques in quantum information theory

Debbie Leung, Caltech & U. Waterloo

roll up our sleeves & prove a few things

Unifying & Simplifying Measurement-based Quantum Computation Schemes


Light, 95% math freemay contain traces of physics

Unifying & Simplifying Measurement-based Quantum Computation Schemes


quant-ph/0404082,0404132Joint work with Panos Aliferis, Andrew Childs, & Michael Nielsen

Hashing ideas from Charles Bennett, Hans Briegel, Dan Browne, Isaac Chuang, Daniel Gottesman, Robert Raussendorf, Xinlan Zhou

Universal QC schemes using only simple measurements:

1WQC:

• Universal entangled initial state• 1-qubit measurements


1) One-way Quantum Computer “1WQC” (Raussendorf & Briegel 00)

: |+i = |0i+|1i,

: controlled-Z

Cluster state:

Can be easily prepared by (1) |+i + controlled-Z, or ZZ, or(2) measurements of stabilizers e.g.

X ZZ

Z

Z

TQC:

• Any initial state (e.g. j000i)• 1&2-qubit measurements

j0i

⋮

j0i

u B==

B==

B==

u B==

u B==


2) Teleportation-based Quantum Computation “TQC” (Nielsen 01, L 01,03)

Basic idea in each box:

Bell

XcZdU

U c,d

1WQC:

• Universal entangled initial state• 1-qubit measurements

TQC:

• Any initial state (e.g. j000i)• 1&2-qubit measurements

j0i

⋮

j0i

u B==

B==

B==

u B==

u B==


1) One-way Quantum Computer “1WQC” (Raussendorf & Briegel 00)

2) Teleportation-based Quantum Computation “TQC” (Nielsen 01, L 01,03)

Rest of talk: 0. Define simulation 1. Review 1-bit-teleportation

Qn: are 1WQC & TQC related & can they be simplified?

Here: derive simplified versions of both using

“1-bit-teleportation” (Zhou, L, Chuang 00)

(simplified version of Gottesman & Chuang 99)

milk

strawberry

strawberry ice-cream & strawberry smoothy

freeze & mixor

mix & freeze

2. Derive intermediate simulation circuits (using much more than measurements) for a universal set of gates

3. Derive measurement-only schemes

Ans: 1WQC = repeated use of the teleportation idea Then a big simplification suggests itself.

Standard model for universal quantum computation :

U

U

0/1

0/1

0/1

U

U5

Un

U

00 : :0

:

:

time

initial state Computation: gates from a universal gate set measure

DiVincenzo 95

Wanted: a notion of “composable” elementwise-simulation

Simulation of components up to known “leftist” Paulis

(input to U), XaZb (arbitrary known Pauli operator)

(c,d) only dependson (a,b,k)

kXaZb

U XcZd

(a,b)

U U

Intended evolution Simulation

U simulates itself

,a,b UXaZb = XcZdU

U Clifford group

e.g.

e.g. U

X,Z: Pauli operators, a,b,c,d {0,1}

U simulates I

,a,b UXaZb = XcZd U Pauli group

e.g.

U

U

0/1

0/1

0/1

U

U5

Un

00 : :0

:

Composing simulations to simulate any circuit :

Simulation of circuit up to known “leftist” Paulis

U

U

0/1

0/1

0/1

U5

Un

00 : :0


XaZb

XaZb :

U


XaZb

U

U

State = (Xa ) (Xa )

U

U

0/1

0/1

0/1

U5

Un

00 : :0


XaZb :

UXaZb

XaZb


State = (Xa ) (Xa )

0/1

0/1

0/1

U5

Un

00 : :0


XaZb

U

U

XaZb :

U

XaZb


State = (Xa ) (Xa )

→ XcZd U2U1

U

U

0/1

0/1

0/1

U

U5

Un

00 : :0


XaZb :

XcZd

XcZd


→ XcZd U2U1

→ Xe Zf U3U2U1

State = (Xa ) (Xa )

U

U

0/1

0/1

0/1

U

U5

Un

00 : :0


:

XaZb

XaZb

XaZb

XaZb


U

U

0/1

0/1

0/1

U

U5

Un

00 : :0


:

XaZb

XaZb

XaZb

XaZb

Propagate errors without affecting the computation. Final measurement outcomes are flipped in a known (harmless) way.


1-bit teleportation

Z-Telepo (ZT)

H c|0i

|i

Zc|i

H

d

c

d c

Teleportation without correction

CNOT:Recall: Pauli’s:

I, X, Z

Hadamard:

H

H

d

X-rtation (XT)

|0i

|i

Xd|iNB. All simulate “I”.

Simulating a universal set of gates: Z & X-rotations

(1-qubit gates) & controlled-Z with mixed resources.

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

= Xa Zc+b eiZ|i

Z-Telep (ZT)

Input state = ei(-1)aZ XaZb |iXa eiZ

c

Z-Telep (ZT)

H c|0i

|i

Zc|i

XaZb|i H|0i

ei(-1)aZ

Simulating a Z rotation eiZ

Xa Zc+b eiZ|i

c

Z-Telep (ZT)

H

H

c

d

|0i

|i

Zc|i

X-Telep (XT)

|0i

|i

Xd|i

Xa Zc+b eiZ|i

XaZb|i

Xa+d Zb eiX|i


H|0i

ei(-1)aZ

Simulating an X rotation eiX

H

d|0i

XaZb|i ei(-1)bX

H

d

X-Telep (XT)

|0i

|i

Xd|i

1 0 0 00 1 0 00 0 1 0 0 0 0 -1

=C-Z:

H

d1

|0i

H

d2

|0i

Xa1Zb1 Xa2Zb2|i

Simulating a C-Z

Xa1+d1Zb1+a2+d2Xa2+d2Zb2+a1+d1 C-Z |i

From simulation with mixed resources to TQC --

QC by 1&2-qubit projective measurements only

c

Xa Zc+b eiZ|i

XaZb|i


H|0i

ei(-1)aZ

c

Xa+a2 Zc+b eiZ|i

XaZb|i


Hei(-1)aZ

An incomplete 2-qubit measurement, followed by a complete measurement on the 1st qubit .

“Xa2”

|0 up to Xa2

jHU

VZ

V†

j

U†XU V†ZVU†ZU k

V†ZkV=

A little fact:

O = measurement of operator O

c

Xa+a2 Zc+b eiZ|i

XaZb|i


“Xa2”

c

Xa+a2 Zc+b eiZ|i

XaZb|i


Hei(-1)aZ

“Xa2”

Xa+d Zb+b2 eiX|i


dXaZb|i ei(-1)bX

“Zb2”

c

Xa+a2 Zc+b eiZ|i

XaZb|i


Hei(-1)aZ

“Xa2”

Xa+d Zb+b2 eiX|i


dXaZb|i ei(-1)bX

“Zb2”

H

d1

|0i

H

d2

|0i

Xa1Zb1 Xa2Zb2|iSimulating a C-Z

Xa1’ Zb1’ Xa2’ Zb2’ C-Z |i

c

Xa+a2 Zc+b eiZ|i

XaZb|i


Hei(-1)aZ

“Xa2”

Xa+d Zb+b2 eiX|i


dXaZb|i ei(-1)bX

“Zb2”

d1d2


H|0i

H|0i Xa1’ Zb1’ Xa2’ Zb2’ C-Z |i

c

Xa+a2 Zc+b eiZ|i

XaZb|i


Hei(-1)aZ

“Xa2”

Xa+d Zb+b2 eiX|i


dXaZb|i ei(-1)bX

“Zb2”

d1d2


Xa1’ Zb1’ Xa2’ Zb2’ C-Z |i

Complete recipe for TQC based on 1-bit teleportation

Aside: universality of 2-qubit meas is immediate!

BellBell

c,d

2-qubit gate to be teleported

4-qubit state to be prepared

d1d2

Previous TQC with full teleportation:

H|0i

H|0i

Simplified TQC with 1-bit teleportation:

2-qubit state to be prepared

jHHZ

j

Z ZX kZk

=

With slight improvements (see quant-ph/0404132):

n-qubit m C-Z up to (m+1)n 1-qubit gates circuit

Sufficient 2m 2-qubit meas 2m+n 1-qubit meas in TQC

Deriving 1WQC-like schemes using gate simulations

obtained from 1-bit teleportation

1WQC:

• Universal entangled initial state• Feedforward 1-qubit measurement

General circuit:

...

Alternating: (1) 1-qubit gates (2) nearest neighbor optional C-Z

General circuit:

...

Rz Rx Rz

Rz Rx Rz

Rz Rx Rz

Rz Rx Rz

Rz Rx Rz

Rz Rx Rz

Rz

Rz

Rz

Rz

Alternating: (1) 1-qubit gates (2) nearest neighbor optional C-Z

Z rotations + optional C-Z – X rotations – Z rotations + optional C-Z – X rotations – ....

simulate these 2 things

Euler-angle decomposition

Xa+d Zb eiX|i


H

d|0i

XaZb|i ei(-1)bX

Adding an optional C-Z right before Z rotations

c1H

|0ic2H

|0i

ei(-1)a1Z

ei(-1)a2Z

Xa1Zb1 Xa2Zb2|i

Xa1 Zb1+a2 k

Xa2 Zb2 +a1 k C-Zk|i Xa1 Zc1+b1+a2 k Xa2 Zc2+b2+a2 k eiZeiZ C-Zk|i

Will derive a method for optional C-Z later : the ability to choose to simulate I or C-Z

|i

optional

c1c2

H

|0iH

|0i

ei(-1)a1Z

ei(-1)a2Z

H

d1

|0i

ei(-1)bX

H

d2

|0i

ei(-1)bX

c1’c2’

H

|0iH

|0i

ei(-1)a1Z

ei(-1)a2Z

H

d1’

|0i

ei(-1)bX

H

d2’

|0i

ei(-1)bX

C-Z+Z rotations –-- X rotations –-- C-Z+Z rotations –-- X rotations ...

Chaining up

optional

|ic1c2

H

|0iH

|0i

ei(-1)a1Z

ei(-1)a2Z

H

d1

|0i

ei(-1)bX

H

d2

|0i

ei(-1)bX

c1’c2’

H

|0iH

|0i

ei(-1)a1Z

ei(-1)a2Z

H

d1’

|0i

ei(-1)bX

H

d2’

|0i

ei(-1)bX

Use

H|0i = |+i, HH=I

H HH

H HH

H

H

H HH

H HH

H

H

= H H


Chaining up

|ic1c2

H

|+iH

|+i

ei(-1)a1Z

ei(-1)a2Z

c1c2

H

|+iH

|+i

ei(-1)a1Z

ei(-1)a2Z

d1

|+i

ei(-1)bX

d2

|+i

ei(-1)bXH

H

optional

d1

|+i

ei(-1)bX

d2

|+i

ei(-1)bXH

H


Chaining up

= |+iLet ,

Then, initial state =

|i

Circuit dependent initial state: 3 qubits, 8 cycles of C-Z + 1-qubit rotations

C-ZZ-rotations X-rotations

H

d1

|0i

H

d2

|0i

Xa1Zb1 Xa2Zb2|i

Recall : simulating a C-Z

Simulating an optional C-Z

Xa1’ Zb1’

Xa2’ Zb2’ C-Z |i

H

d1

|0i

H

d2

|0i

Recall : simulating a C-Z

Hd1

|0i

Hd2

|0i

1. Redrawing the 2nd input to the bottom:


2. Use symmetry:

Hd1

|0i

Hd2

|0i

1. Redrawing the 2nd input to the bottom:

Just measures the parity of the 2 qubits

It is equal to


j

Xj

j

2. Use symmetry:

H|0i

H|0i d2d1


Just measures the parity of the 2 qubits

It is equal to

j

Xj

j

H|0i

H|0i d2d1


H|0i

H|0i d2d1

3. Use

H =

= H H


H|0i

H|0i d2d1H

H


3. Use

H =

= H H

3. Use H|0iH|0i

H

H

H|0i

H|0i=


H|0i

H|0i d2d1H

H

3. Use H|0iH|0i

H

H

H|0i

H|0i=


H|0i

H|0i d2d1

3. Use H|0i =|+i,

|+i|+i d2

d1“Remote C-Z” : Cousin of the remote CNOT by Gottesman98

H|0i

H|0i d2d1


|+i|+i

If one measures along {|0i, |1i}, the C-Zs labeled by ①② only acts like Zd1 Zd2 – simulating identity instead!

d1d2

①

②


|+i|+i d2

d1“Remote C-Z” : Cousin of the remote CNOT by Gottesman98

Simulating an optional C-Z, summary:

|+i|+i d2

d1 |+i|+i

d1d2

simulates

To do the C-Z: To skip the C-Z:

Simulating an optional C-Z, summary:

|+i|+i d2

d1 |+i|+i

d1d2

simulates

To do the C-Z: To skip the C-Z:

also simulates Do: Skip:

|+i Y |+i Z

up to Z-rotations

Universal Initial state 3 qubits, 8 cycles

Starting from the cluster state

measure in Z basis

Universal Initial state 3 qubits, 8 cycles

Starting from the cluster state

measure in Z basis

Other universal initial state with other methods for optional C-Z

Other universal initial state with other methods for optional C-Z

Summary:

Unified derivations, using 1-bit teleportation,

for 1WQC & TQC + simplifications

Details in quant-ph/0404082,0404132

...

Related results by Perdrix & Jorrand, Verstraete & Cirac

but perhaps you don’t need to see them, you only need to remember what is a simulation (milk), what 1-bit teleportation does (strawberry), and the rest (mix/freeze) comes naturally.

Summary:

1-bit teleportation has been used for systematic derivation of

simplified constructions of fault tolerant gates.

We have seen a similar use in deriving measurement-based QC.

It leads to the remote C-Z/CNOT and programmable gate-array.

Does it has a special role in quantum information theory?

Open issues

When it is already so simple ?

Further optimizations?

Open issues

When it is already so simple ?

Practically, the most interesting problems are likely to involve a

mixture of resources, not just measurements.

Measurement-based QC is most important as a conceptual tool.

“Strawberry milkshake” taste much better with banana in it !

Open issues mixture of resources, not just measurements.

Running 1WQC in linear optics Nielsen 04, Drowne & Rudolph 04

Problem in linear optics:

- C-Z difficult

- C-Z probabilistically by teleportation trick Knill, Laflamme, Milburn01

Idea: Applying faulty C-Z to the data is expensive & failures are

painful to repair. Instead, apply C-Z to build a cluster/graph state

followed by 1-qubit measurements. Faulty C-Zs percolates the

cluster state, but cheap to repair.

Qn: optimize construction.

Tradeoff between different methods to protect against

percolations, e.g. good expensive C-Z vs redundant coding

Qn: threshold under linear optics model? (1WQC: Raussendorf PhD thesis, Nielsen & Dawson 04)

Open issues (what else to add to pure strawberry milkshake?)

no threshold without fresh ancillas and interaction in 1WQC

What are reasonable models for such resources ?

What are reasonable models for the noise?

Again, will be more experimentally motivated.

e.g. Photons? Trapped ions? Quantum dots?

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roll up our sleeves & prove a few things

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