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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 Debbie Leung, Caltech & U. Waterloo. Light, 95% math free - PowerPoint PPT Presentation
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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
Debbie Leung, Caltech & U. Waterloo
Light, 95% math freemay contain traces of physics
Unifying & Simplifying Measurement-based Quantum Computation Schemes
Debbie Leung, Caltech & U. Waterloo
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
Universal QC schemes using only simple 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==
Universal QC schemes using only simple measurements:
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==
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)
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
Composing simulations to simulate any circuit :
XaZb
XaZb :
U
Simulation of circuit up to known “leftist” Paulis
XaZb
U
U
State = (Xa ) (Xa )
U
U
0/1
0/1
0/1
U5
Un
00 : :0
Composing simulations to simulate any circuit :
XaZb :
UXaZb
XaZb
Simulation of circuit up to known “leftist” Paulis
State = (Xa ) (Xa )
0/1
0/1
0/1
U5
Un
00 : :0
Composing simulations to simulate any circuit :
XaZb
U
U
XaZb :
U
XaZb
Simulation of circuit up to known “leftist” Paulis
State = (Xa ) (Xa )
→ XcZd U2U1
U
U
0/1
0/1
0/1
U
U5
Un
00 : :0
Composing simulations to simulate any circuit :
XaZb :
XcZd
XcZd
Simulation of circuit up to known “leftist” Paulis
→ XcZd U2U1
→ Xe Zf U3U2U1
State = (Xa ) (Xa )
U
U
0/1
0/1
0/1
U
U5
Un
00 : :0
Composing simulations to simulate any circuit :
:
XaZb
XaZb
XaZb
XaZb
Simulation of circuit up to known “leftist” Paulis
U
U
0/1
0/1
0/1
U
U5
Un
00 : :0
Composing simulations to simulate any circuit :
:
XaZb
XaZb
XaZb
XaZb
Propagate errors without affecting the computation. Final measurement outcomes are flipped in a known (harmless) way.
Simulation of circuit up to known “leftist” Paulis
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
Simulating a Z rotation eiZ
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
Simulating a Z rotation eiZ
H|0i
ei(-1)aZ
c
Xa+a2 Zc+b eiZ|i
XaZb|i
Simulating a Z rotation eiZ
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
Simulating a Z rotation eiZ
“Xa2”
c
Xa+a2 Zc+b eiZ|i
XaZb|i
Simulating a Z rotation eiZ
Hei(-1)aZ
“Xa2”
Xa+d Zb+b2 eiX|i
Simulating an X rotation eiX
dXaZb|i ei(-1)bX
“Zb2”
c
Xa+a2 Zc+b eiZ|i
XaZb|i
Simulating a Z rotation eiZ
Hei(-1)aZ
“Xa2”
Xa+d Zb+b2 eiX|i
Simulating an X rotation eiX
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
Simulating a Z rotation eiZ
Hei(-1)aZ
“Xa2”
Xa+d Zb+b2 eiX|i
Simulating an X rotation eiX
dXaZb|i ei(-1)bX
“Zb2”
d1d2
Xa1Zb1 Xa2Zb2|iSimulating a C-Z
H|0i
H|0i Xa1’ Zb1’ Xa2’ Zb2’ C-Z |i
c
Xa+a2 Zc+b eiZ|i
XaZb|i
Simulating a Z rotation eiZ
Hei(-1)aZ
“Xa2”
Xa+d Zb+b2 eiX|i
Simulating an X rotation eiX
dXaZb|i ei(-1)bX
“Zb2”
d1d2
Xa1Zb1 Xa2Zb2|iSimulating a C-Z
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
Simulating an X rotation eiX
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
C-Z+Z rotations –-- X rotations –-- C-Z+Z rotations –-- X rotations ...
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
C-Z+Z rotations –-- X rotations –-- C-Z+Z rotations –-- X rotations ...
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:
Simulating an optional C-Z
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
Simulating an optional C-Z
j
Xj
j
2. Use symmetry:
H|0i
H|0i d2d1
Simulating an optional C-Z
Just measures the parity of the 2 qubits
It is equal to
j
Xj
j
H|0i
H|0i d2d1
Simulating an optional C-Z
H|0i
H|0i d2d1
3. Use
H =
= H H
Simulating an optional C-Z
H|0i
H|0i d2d1H
H
Simulating an optional C-Z
3. Use
H =
= H H
3. Use H|0iH|0i
H
H
H|0i
H|0i=
Simulating an optional C-Z
H|0i
H|0i d2d1H
H
3. Use H|0iH|0i
H
H
H|0i
H|0i=
Simulating an optional C-Z
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
Simulating an optional C-Z
|+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
Simulating an optional C-Z
|+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?