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Applications of randomized techniques in quantum information theoryDebbie Leung, Caltech & U. Waterloo

roll up our sleeves

& prove a few things

Unifying & Simplifying Measurement-based Quantum Computation Schemes Debbie Leung, Caltech & U. Waterloo

Light, 95% math free

may contain traces of physics

Unifying & Simplifying Measurement-based Quantum Computation Schemes Debbie Leung, Caltech & U. Waterlooquant-ph/0404082,0404132Joint work with Panos Aliferis, Andrew Childs, & Michael NielsenHashing ideas from Charles Bennett, Hans Briegel, Dan Browne, Isaac Chuang, Daniel Gottesman, Robert Raussendorf, Xinlan Zhou

: |+ Schemes i = |0i+|1i

,

Z

: controlled-Z

Z

X

Z

Z

Universal QC schemes using only simple measurements:

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

1WQC:

• Universal entangled initial state

• 1-qubit measurements

Cluster state:

Can be easily prepared by

(1) |+i + controlled-Z, or ZZ, or

(2) measurements of stabilizers

e.g.

Basic idea in each box: Schemes

j

U

c,d

Bell

j0i

⋮

j0i

XcZdUj

=

=

B

=

=

=

=

u

B

B

=

=

=

=

u

u

B

B

Universal QC schemes using only simple measurements:

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

L 01,03)

TQC:

• Any initial state (e.g. j00L0i)

• 1&2-qubit measurements

j Schemes 0i

⋮

j0i

=

=

B

=

=

=

=

u

B

B

=

=

=

=

u

u

B

B

Universal QC schemes using only simple measurements:

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

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

L 01,03)

1WQC:

• Universal entangled initial state

• 1-qubit measurements

TQC:

• Any initial state (e.g. j00L0i)

• 1&2-qubit measurements

strawberry ice-cream & strawberry smoothy Schemes

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)

Ans: 1WQC = repeated use of the teleportation idea

Then a big simplification suggests itself.

Rest of talk: 0. Define simulation

1. Review 1-bit-teleportation

milk

strawberry

2. Derive intermediate simulation circuits (using much more

than measurements) for a universal set of gates

3. Derive measurement-only schemes

freeze & mixor

mix & freeze

00 Schemes

:

:

0

0/1

U5

U3

U1

0/1

:

U4

:

Un

U2

0/1

time

Standard model for universal quantum computation :

DiVincenzo 95

Computation:

gates from a universal gate set

initial state

measure

Wanted: a notion of “composable” elementwise-simulation

Schemes j

XaZb

k

j

XcZd

U

(a,b)

e.g.

U simulates itself

j,a,b UXaZbj = XcZdUj

U Clifford group

Simulation of componentsup to known “leftist” Paulis

e.g. U

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

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

Intended evolution

Simulation

j

U j

U

(c,d) only

depends

on (a,b,k)

U simulates I

j,a,b UXaZbj = XcZdj U Pauli group

e.g.

Simulation of circuit Schemes up to known “leftist” Paulis

Composing simulations to simulate any circuit :

00

:

:

0

0/1

U5

U3

U1

0/1

:

Un

U2

0/1

Simulation of circuit Schemes up to known “leftist” Paulis

Composing simulations to simulate any circuit :

00

:

:

0

XaZb

0/1

U5

U3

U1

U1

XaZb

0/1

:

Un

XaZb

U2

U2

0/1

State = (Xa 0) (Xa 0)

Simulation of circuit Schemes up to known “leftist” Paulis

Composing simulations to simulate any circuit :

00

:

:

0

XaZb

0/1

U5

U3

U1

XaZb

0/1

:

Un

XaZb

U2

0/1

State = (Xa 0) (Xa 0)

U Schemes 1

U2

Simulation of circuitup to known “leftist” Paulis

Composing simulations to simulate any circuit :

00

:

:

0

XaZb

0/1

U5

U3

XaZb

0/1

:

Un

XaZb

0/1

State = (Xa 0) (Xa 0)

→XcZdU2U10 0

Simulation of circuit Schemes up to known “leftist” Paulis

Composing simulations to simulate any circuit :

00

:

:

0

XcZd

0/1

U5

U3

U1

XcZd

0/1

:

Un

XaZb

U2

0/1

State = (Xa 0) (Xa 0)

→XcZdU2U10 0

→Xe ZfU3U2U10 0

Simulation of circuit Schemes up to known “leftist” Paulis

Composing simulations to simulate any circuit :

00

:

:

0

XaZb

0/1

U5

U3

U1

XaZb

0/1

XaZb

:

Un

XaZb

U2

0/1

Simulation of circuit Schemes up to known “leftist” Paulis

Composing simulations to simulate any circuit :

00

:

:

0

XaZb

0/1

U5

U3

XaZb

U1

0/1

XaZb

:

Un

XaZb

U2

0/1

Propagate errors without affecting the

computation. Final measurement outcomes

are flipped in a known (harmless) way.

1-bit teleportation Schemes

H Schemes

c

d

d

c

X-rtation (XT)

|i

d

|0i

H

Xd|i

Z-Telepo (ZT)Teleportation without correction

|i

H

c

|0i

Zc|i

NB. All simulate “I”.

CNOT:

Hadamard:

Recall:

Pauli’s:

I, X, Z

H

Simulating a universal set of gates: Z & X-rotations ( Schemes 1-qubit gates) & controlled-Z with mixed resources.

Goal: perform Z rotation Schemes eiqZ

X Schemes aZb|i

ei(-1)aqZ

H

c

|0i

Zcei(-1)aqZ XaZb|i

= Xa Zc+beiqZ|i

Z-Telep (ZT)Goal: perform Z rotation eiqZ

|i

H

c

|0i

Zc|i

XaeiqZ

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

Z-Telep (ZT) Schemes

Simulating a Z rotation eiqZ

XaZb|i

|i

ei(-1)aqZ

H

c

H

c

|0i

Xa Zc+beiqZ|i

|0i

Zc|i

Z-Telep (ZT) Schemes

Simulating a Z rotation eiqZ

XaZb|i

|i

ei(-1)aqZ

H

c

H

c

|0i

Xa Zc+beiqZ|i

|0i

Zc|i

X-Telep (XT)

Simulating an X rotation eiqX

XaZb|i

ei(-1)bqX

|i

d

d

|0i

Xa+d ZbeiqX|i

|0i

H

H

Xd|i

Simulating a C-Z Schemes

Xa1Zb1 Xa2Zb2|i

d1

d2

|0i

H

|0i

H

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

X-Telep (XT)

|i

d

|0i

H

Xd|i

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 -1

C-Z:

=

From simulation with mixed resources to TQC -- Schemes

QC by 1&2-qubit projective measurements only

U Schemes

H

j

V†

V

=

j

U†ZU

U†XU V†ZV

k

V†ZkV

O

= measurement of operator O

Z

Simulating a Z rotation eiqZ

XaZb|i

ei(-1)aqZ

H

c

Xa+a2 Zc+beiqZ|i

“Xa2”

|0 up to Xa2

An incomplete 2-qubit measurement, followed

by a complete measurement on the 1st qubit .

A little fact:

Simulating a Z rotation Schemes eiqZ

XaZb|i

ei(-1)aqZ

H

c

Xa+a2 Zc+beiqZ|i

“Xa2”

Simulating an X rotation eiqX

XaZb|i

ei(-1)bqX

d

Xa+d Zb+b2eiqX|i

“Zb2”

X Schemes a1’ Zb1’ Xa2’ Zb2’ C-Z|i

Simulating a Z rotation eiqZ

XaZb|i

ei(-1)aqZ

H

c

Xa+a2 Zc+beiqZ|i

“Xa2”

Simulating an X rotation eiqX

XaZb|i

ei(-1)bqX

d

Xa+d Zb+b2eiqX|i

“Zb2”

Xa1Zb1 Xa2Zb2|i

Simulating a C-Z

d1

d2

|0i

H

|0i

H

X Schemes a1’ Zb1’ Xa2’ Zb2’ C-Z|i

Simulating a Z rotation eiqZ

XaZb|i

ei(-1)aqZ

H

c

Xa+a2 Zc+beiqZ|i

“Xa2”

Simulating an X rotation eiqX

XaZb|i

ei(-1)bqX

d

Xa+d Zb+b2eiqX|i

“Zb2”

Xa1Zb1 Xa2Zb2|i

Simulating a C-Z

d1

d2

|0i

H

|0i

H

X Schemes a1’ Zb1’ Xa2’ Zb2’ C-Z|i

Simulating a Z rotation eiqZ

XaZb|i

ei(-1)aqZ

H

c

Xa+a2 Zc+beiqZ|i

“Xa2”

Complete recipe

for TQC based on

1-bit teleportation

Simulating an X rotation eiqX

XaZb|i

ei(-1)bqX

d

Xa+d Zb+b2eiqX|i

“Zb2”

Xa1Zb1 Xa2Zb2|i

Simulating a C-Z

d1

d2

H Schemes

H

Z

j

=

j

X

Z Z

k

Zk

Aside: universality of 2-qubit meas is immediate!

Previous TQC with full teleportation:

Bell

Bell

c,d

4-qubit state to be prepared

2-qubit gate to be teleported

Simplified TQC with 1-bit teleportation:

d1

d2

|0i

H

|0i

H

2-qubit state

to be prepared

With slight improvements (see quant-ph/0404132): Schemes n-qubitm C-Z up to (m+1)n1-qubit gatescircuitSufficient 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: from 1-bit teleportation

...

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

R from 1-bit teleportationz

Rz

Rz

Rz

Rz

Rz

Rx

Rx

Rx

Rx

Rx

Rx

Rz

Rz

Rz

Rz

Rz

Rz

General circuit:Rz

Rz

Rz

Rz

...

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

Euler-angle

decomposition

Z rotations + optional C-Z – X rotations –

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

simulate these 2 things

X from 1-bit teleportationa1 Zb1+a2k

Xa2 Zb2 +a1k

C-Zk|i

Xa1 Zc1+b1+a2k

Xa2 Zc2+b2+a2k

eiq1Zeiq2ZC-Zk|i

Simulating an X rotation eiqX

XaZb|i

ei(-1)bqX

d

|0i

Xa+d ZbeiqX|i

H

Adding an optional C-Z right before Z rotations

ei(-1)a1q1Z

H

c1

Xa1Zb1 Xa2Zb2|i

ei(-1)a2q2Z

c2

H

|0i

|0i

Will derive a method for optional C-Z later :

the ability to choose to simulate I or C-Z

Chaining up from 1-bit teleportation

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

c1

c1’

c2’

c2

ei(-1)bqX

d1

ei(-1)bqX

d2

|0i

H

|0i

H

optional

ei(-1)a1q1Z

H

|i

ei(-1)a2q2Z

H

|0i

|0i

ei(-1)a1q1Z

H

ei(-1)a2q2Z

H

|0i

|0i

ei(-1)bqX

d1’

ei(-1)bqX

d2’

|0i

H

|0i

H

Chaining up from 1-bit teleportation

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

=

c1

c1’

H

H

c2

c2’

optional

Use

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

ei(-1)a1q1Z

H

|i

ei(-1)a2q2Z

H

|0i

H

H

|0i

H

H

ei(-1)bqX

H

H

d1

H

H

ei(-1)bqX

d2

|0i

H

|0i

ei(-1)a1q1Z

H

H

ei(-1)a2q2Z

H

|0i

H

H

|0i

H

H

ei(-1)bqX

d1’

H

H

H

ei(-1)bqX

H

d2’

|0i

H

|0i

H

Chaining up from 1-bit teleportation

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

c1

c1

c2

c2

|i

= |+i

Let

,

optional

Then, initial state =

ei(-1)a1q1Z

H

|i

ei(-1)a2q2Z

H

|+i

|+i

ei(-1)bqX

H

d1

H

ei(-1)bqX

d2

|+i

|+i

ei(-1)a1q1Z

H

ei(-1)a2q2Z

H

|+i

|+i

ei(-1)bqX

d1

H

ei(-1)bqX

H

d2

|+i

|+i

Circuit dependent initial state: from 1-bit teleportation

3 qubits, 8 cycles of C-Z + 1-qubit rotations

C-Z

Z-rotations

X-rotations

Simulating an optional C-Z from 1-bit teleportation

Recall : simulating a C-Z

Xa1Zb1 Xa2Zb2|i

d1

d2

|0i

H

Xa1’ Zb1’

Xa2’ Zb2’ C-Z|i

|0i

H

Simulating an optional C-Z from 1-bit teleportation

Recall : simulating a C-Z

d1

d2

|0i

H

|0i

H

1. Redrawing the 2nd input to the bottom:

d1

|0i

H

|0i

H

d2

X from 1-bit teleportationj

Simulating an optional C-Z

2. Use symmetry:

Just measures the parity of the 2 qubits

j

j

It is equal to

1. Redrawing the 2nd input to the bottom:

d1

|0i

H

|0i

H

d2

X from 1-bit teleportationj

d2

d1

Simulating an optional C-Z

2. Use symmetry:

Just measures the parity of the 2 qubits

j

j

It is equal to

|0i

H

|0i

H

H from 1-bit teleportation

H

|0i

|0i

H

H

|0i

H

H

d1

|0i

d2

H

H

Simulating an optional C-Z

3. Use

|0i

H

=

|0i

H

H from 1-bit teleportation

H

|0i

|0i

H

H

Simulating an optional C-Z

3. Use

|0i

H

=

|0i

H

|0i

H

d1

|0i

d2

H

Simulating an optional C-Z from 1-bit teleportation

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

“Remote C-Z” :

Cousin of the remote

CNOT by Gottesman98

|+i

d1

|+i

d2

|0i

H

d1

|0i

d2

H

Simulating an optional C-Z from 1-bit teleportation

“Remote C-Z” :

Cousin of the remote

CNOT by Gottesman98

|+i

d1

|+i

d2

If one measures along {|0i,|1i},

the C-Zs labeled by ①② only acts

like Zd1 Zd2 – simulating identity

instead!

①

②

|+i

d1

|+i

d2

|+ from 1-bit teleportationi

d1

|+i

d2

|+i

d1

|+i

d2

Simulating an optional C-Z, summary:

simulates

To do the C-Z:

To skip the C-Z:

|+ from 1-bit teleportationi

d1

|+i

d2

|+i

d1

|+i

|+i

Y

Z

|+i

d2

Simulating an optional C-Z, summary:

simulates

To do the C-Z:

To skip the C-Z:

Do:

Skip:

also simulates

up to Z-rotations

Universal Initial state from 1-bit teleportation

3 qubits, 8 cycles

Starting from the cluster state from 1-bit teleportation

measure in Z basis

Universal Initial state from 1-bit teleportation

3 qubits, 8 cycles

Starting from the cluster state from 1-bit teleportation

measure in Z basis

Other universal initial state with from 1-bit teleportationother methods for optional C-Z

Other universal initial state with from 1-bit teleportationother methods for optional C-Z

Summary: from 1-bit teleportationUnified derivations, using 1-bit teleportation, for 1WQC & TQC + simplificationsDetails in quant-ph/0404082,0404132 ...

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.

Related results by Perdrix & Jorrand, Verstraete & Cirac

Summary: from 1-bit teleportation1-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 from 1-bit teleportation When it is already so simple ?Further optimizations?

Open issues from 1-bit teleportation 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 from 1-bit teleportation mixture of resources, not just measurements. Running 1WQC in linear optics Nielsen 04, Drowne & Rudolph 04Problem in linear optics: - C-Z difficult - C-Z probabilistically by teleportation trick Knill, Laflamme, Milburn01Idea: 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 from 1-bit teleportation (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?