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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

Universal QC schemes using only simple measurements:

: |+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:

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

j0i

⋮

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

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

:

:

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

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 circuitup 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 circuitup 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 circuitup 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)

U1

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 circuitup 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 circuitup 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 circuitup 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

H

c

d

d

c

X-rtation (XT)

|i

d

|0i

H

Xd|i

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

Goal: perform Z rotation eiqZ

Goal: perform Z rotation eiqZ

|i

H

c

|0i

Zc|i

XaZb|i

ei(-1)aqZ

H

c

|0i

Zcei(-1)aqZ XaZb|i

= Xa Zc+beiqZ|i

Goal: perform Z rotation eiqZ

|i

H

c

|0i

Zc|i

XaeiqZ

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

Simulating a Z rotation eiqZ

XaZb|i

|i

ei(-1)aqZ

H

c

H

c

|0i

Xa Zc+beiqZ|i

|0i

Zc|i

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

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 --

QC by 1&2-qubit projective measurements only

Simulating a Z rotation eiqZ

XaZb|i

ei(-1)aqZ

H

c

|0i

Xa Zc+beiqZ|i

U

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 eiqZ

XaZb|i

c

Xa+a2 Zc+beiqZ|i

“Xa2”

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”

Xa1’ 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

Xa1’ 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

Xa1’ 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

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): 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

...

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

Rz

Rz

Rz

Rz

Rz

Rz

Rx

Rx

Rx

Rx

Rx

Rx

Rz

Rz

Rz

Rz

Rz

Rz

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

Xa1 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

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

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

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

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

C-Z

Z-rotations

X-rotations

Simulating an optional C-Z

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

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

Xj

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

Xj

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

d2

d1

Simulating an optional C-Z

|0i

H

|0i

H

3. Use

=

H

H

=

H

d2

d1

Simulating an optional C-Z

|0i

H

|0i

H

3. Use

=

H

H

=

H

|0i

H

H

d1

|0i

d2

H

H

Simulating an optional C-Z

H

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

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

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

“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

|+i

d1

|+i

d2

|+i

d1

|+i

d2

Simulating an optional C-Z, summary:

simulates

To do the C-Z:

To skip the C-Z:

|+i

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

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 + 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: 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 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 (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?