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1
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Notes on Quantum Computation
E.G. VillaniSTFC
Rutherford Appleton Laboratory
2
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
OutlineOutline
• Quantum Computation introduction
• Algorithms examples
• Quantum Computation Technology
• Conclusions
3
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Introduction notesIntroduction notes
Topics:Quantum information and computation theoryImplementation using different technologies
Topological Quantum computingQuantum magnetism
… 25th Winter School in Theoretical Physics
Institute for Advanced StudiesJerusalem 2007
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Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Jerusalem notesJerusalem notes
Unified (east and west) 1967 Capital city of Israel from 1980Around 800,00 inhabitants : approx 70% Jewish, 29% Muslim, 1% ChristianMostly religious: ‘secular’ population in constant decline Old city around 1km2 :one of the oldest cities in the world
Christian
ArmenianJewish
Moslem
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Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Jerusalem notesJerusalem notes
Dome of the ‘Rock’ 7th century AC
Holy SepulchreInitially built by Constantine 4th
Century BC
Western WallRetaining wall for the 2nd
Temple , 1st century BC
6
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
The general process of computation can be described as an operation performed on initial information and the reading out of the results:
Computation –introduction -
Input OutputOp
Computation can be performed in classical domain using analogue or digital blocks
7
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Classical Computation – analogue -
Analogue computation example: PID controller
dtVdt
dVVV in
ininout
sKGsH
sKGsRsC
1
8
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Analogue Computation – analogue -
Analogue Sallen Key 5th order low pass filter
9
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Classical computation – digital -
Digital Sallen Key 5th order low pass filter IIR implementation
jnyjdknxkcny
knxkcny
10
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Quantum limit
Moore’s Law (1965): Exponential size shrinking of electronic devices -> atomic limit will be approached around 2015
11
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
To simulate a probabilistic system (QM system) with a computer requires an exponential increase of resources (i.e. gates, time)
The simulation of a probabilistic system (QM system) could be done more efficiently with a probabilistic (QM) machine: Quantum Computer
1...111...1...0000...000221 Nccct
Quantum Computing –introduction -
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Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
computation in the Quantum domain
10 10
10 10
10 10
operatorinput output
A composite system of N Quantum bits (Qubits) is the input
An unitary operator is applied
A non unitary operator is applied to perform measurement
Quantum Computing –introduction -
13
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Can be depicted as a point on the surface of a Bloch sphere:
In theory a Qubit can store infinite amount of information,
conserved during evolution. Measurement yields only one of the
two values.
1,102
1
2
010
A Quantum bit (qubit) describes the states of each individual two-level systems. In the computational bases:
12
sin02
cos
ie
Quantum Information –introduction -
Quantum information: qubits
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Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
An input state vector of non interacting qubits can be written in the form:
NN ...21
E.g.:
nnin 2121
ii
i
iHiHH 1,0, 22
In a Quantum computer a unit vector in a Hilbert space H describes the initial state of the
system
221214 ,,11002
1 with
Quantum Information
A state vector of interacting qubits consists of entangled states
15
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
In a more general case, a state of n qubits can be found in a
mixed state:
ii
k
iip
1
The density operator used to describe the state of a subsystem, by tracing out the unwanted system
UUUUp ii
k
ii
1
12212121 bbaabbaaTrTr BAB
BA
Quantum Information
16
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Quantum computing –introduction -
To perform a Quantum computation an unitary operation is performed on the qubits system:
00exp UHdti
out
nnn fU 212121
The generator Hamiltonian H has to generate this evolution according to Schrodingers’ equation:
tHtdt
di
The Hamiltonian has to be found for a specific operation U:
Quantum processing: Evolution
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Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Quantum computing
If U unitary a solution for H always exists.
Quantum computers can simulate any classical deterministic (Toffoli gate) and probabilistic (Hadamard gate) functions (i.e. they can perform any computations that
a classical computer does)
If classical f not reversible (like universal Boolean classical gates) it can be made reversible by adding extra information. An unitary quantum equivalent can then be built
E.g. Toffoli gate can be used to make any irreversible classical function reversible. Reversible classical gate (in principle) heatless.
18
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
2, iiii
i ttt
For a given orthonormal bases of HA it is possible to perform Von Neumann measurement:N distinguishable states input to an apparatus that perform a non unitary operation.
Quantum result: Measurement
2
jjjTr
ii i i
Quantum computing
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Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Quantum computing – general process -
General Quantum circuit model
iini i AA
opinput output
0
0
0
U
in
UUTr inBoutin 0...0000...000
iini
iin AA
Superoperator: Quantum operation acts on an inputdensity operator plus ancillary register
If unchanged dimension of H
20
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Quantum gates
A Quantum gate U rotates the Bloch vector on the Bloch sphere
01
10X
1
0
X
11
11
2
1
H
1
0
An unitary 1-qubit Quantum gate U can be written in terms of rotations around non parallel axes of the
Bloch sphere using Pauli gates :
lml
i RRReU
2
01
10 Xi
x eRX
2
0
0 Yi
y eRi
iY
2
10
01 Zi
z eRZ
Quantum computing
21
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
THG ,
Universal Quantum gates
An universal set of Quantum gates allows description with arbitrary accuracy of n-qubit unitary operator
(i.e. equivalent to classical NAND/NOR)
An universal set of Quantum gate for 1-qubit operator
11
11
2
1
H
8
8
1
1
i
i
e
eT
An universal set of Quantum gate for n-qubits operator is obtained by any 2-qubit entangling gate with an universal set for 1-qubit
nTHCNOTGn ,,,
0100
1000
0010
0001
CNOT
Efficiency of approximation ( number of gates) using G1(includes inverses)
1log,max, cOVUVU
Quantum computing
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Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Quantum measurement
V. Neumann measurement
2
jjjTr
V. Neumann measurement with respect to any orthonormal basis
jj j jU
iij
H
Xj
Example: Computational basis to Bell basis
112
100
2
100
112
100
2
110
102
101
2
101
102
101
2
111
Quantum computing
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Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Quantum computing
Quantum communication - teleportation
Bell
ZX
Using two classical bits , it is possible to send the state of a qubit: quantum state transmitted using classical channels!
112
100
2
100
Alice
Bob XZZX 1110010000 2
1
2
1
2
1
2
1
a
b
abab ZXU
Quantum teleportation useful to implement 2-Qubit gates First Quantum teleportation experimentally achieved using photons 1998
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Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Quantum computing – algorithms -
First Quantum algorithm- Deutsch Jozsa Oracle
jj j jU
jj j jΣ ∆
Quantum superposition and interference
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Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Deutsch Quantum algorithm
Problem :
1,01,0: f
To determine if f is constant or balanced.Classically: 2 queries
Deutsch algorithm: 1 query
xfyxyxU f :
Define the reversible mapping :
2
10
10 ff
Input an eigenstate to the target qubit of an operator and associate the eigenvalue with the control register
2
101
xf
xfU
x x
x
xfU
x
y xfyx
xfU
y
x
xfyx
x
Quantum computing – algorithms -
26
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
xfU
0
2
10
H H
0 1 2 3
2
1000
2
101
2
1
2
100
2
11
2
101
2
1
2
100
2
1 10
2
ff
2
10101 0
3 fff
Simultaneous computation
result
Deutsch Quantum algorithm
Quantum computing – algorithms -
27
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
1,01,0: nf
To determine if f is constant or balanced.Classically: 2n-1+1queries
Deutsch algorithm: 1 queryExponential increase in efficiency
xfU
0
2
10
H H
0 1 2 3
0 H H
0 H H
2
1000
n
nx
nx
1,01
2
1
2
101
2
1
1,02
nx
xf
nx
2
101
2
1
1,0 1,03 z
n nz x
zxxf
n
Problem :
Deutsch Jozsa Quantum algorithm
Quantum computing – algorithms -
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Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Quantum algorithm – Quantum Fourier TransformProblem: integer factorization
Split odd- non prime power N
Orders r of integers A co-prime with N
Sampling estimates to random integer multiple 1/n
yexQFTn
n
y
yx
i
n
12
0
22
2
1:
Order of random element in ZN
NNNO loglogloglogloglog 2Quantum complexity
NNOe logloglog Classical complexity
Shor’s algorithm for N factoring could compute 100s digits in seconds
Factorization believed to be NP problem but not demonstrated
xaU
n0 QFT QFT-1
1
NsasU a mod:
Quantum computing – algorithms -
29
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
R(ivest)S(hamir)A(dleman) cryptosystem
Alice BobN,E
P,Q : N = PQ
E: GCD(E,(P-1)(Q-1)=1M
MEmod N
(ME)E-1mod N =M
E-1mod (P-1)(Q-1)
Difficult to factor large numbers: classically ~ weeks for 100 digits N :doubling the digits implies a factorization ~ 106 years!
Quantum computing – algorithms -
30
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Quantum algorithms remarks
Acyclic quantum gate arrays can compute in polynomial time any function computable in polynomial time by CTM .
10 10
10 10
10 10
opinput output
It is not known yet if many NP classical problems can become P using Quantum Computing
QFT (Shor’s algortithm)Deutsch Jozsa algorithm
Grover’s search algorithm( N vs sqrt(N))Simon’s algorithm
Quantum computing – algorithms -
31
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
1-qubit cyclic gate represented by U(2) group
Any P problem can be mapped onto a acyclic quantum gateCyclical quantum gate still not investigated:
Compactness Phase delay
2-qubit cyclic gate represented by U(2) group
11,01
11,10
Quantum computing – further algorithms -
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Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Perturbation of a 2-qubit cyclic network via C-NOT gate
Simplified form for Quantum Gates (i.e. FIR vs IIR in DF)Phase delay may lead to instability
Quantum oscillator!
Evolution of a cyclic network after n cycles:
In U eigenbases:
Simplified QFT
Quantum computing – further algorithms -
33
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Quantum Computing – implementation techniques
Technical issues for building a Quantum Computer (DiVincenzo 2000)
Reliable representation of Quantum Information
(scalable number of Qubits )
Setting of initial state of qubits
Quantum gates reliable (decoherence)
Readout
Quantum decoherence
Interaction with environment ( I.e. partial measurement operated by
the environment)
Errors due to decoherence can be recovered, if error rate is
around 10-3/ 10-4 (Aharonov 1998)
Quantum error correction algorithms are effective if operations
are performed 10+3/ 10+4 faster than decoherence time
Alternative proposed solution to decoherence and error problem is
topological quantum computation
Several proposed solution for quantum
computation
34
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Implementation techniques – Quantum Dots
Quantum well of Si-Ge for bi-dimensional confinement of
electrons, top gates for lateral confinement
Qubits : spins of individual electrons in quantum dots
Orbital coherence time << Spin coherence time (>100ns in
2DEG @ T=5K in GaAs)
Self-assembled Quantum dots using strained epitaxial
growth (i.e. Stranski-Krastanov process, growth of
material on substrate not lattice matched)
10’s nm scale
No nanothechnology required (etching, implanting)
No contamination
Non uniformity in size and position
Quantum dots using lithography :
100nm spacing
Good reproducibility
Nanotechnology required
Contamination
InAs/GaAs SA QD grown with MBE
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Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Qubits : spins of individual electrons in quantum dots
For universal set G, coupling J of spins (qubits) needed
Quantum operation performed
by acting on gate voltages (2-qubit) to control J
ESR (electron spin resonance) ( 1-qubit phase rotation)
Dv/v<10^-6 only at low frequency…
21 SStJtH s
swsw
t
sUUtU
tHitU 2
10 :exp
RO using Spin to Q technique
Implementation techniques – Quantum Dots
e- e- AC
SEM of double Q-dot device
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Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Implementation techniques – Ion traps -
Ions are confined in free UHV space using electromagnetic
field (Paul trap)
Qubits : ground and excited state level or hyperfine levels
Very long decoherence time
Initial state by optical pumping
Measurement using laser pulses coupled to one of the qubit
states: emitted photons read using CCD camera
Not easily scalable
Chip size planar Ions Trap: 6+6 traps of Mg on a flat
alumina surface
Field applied through gold electrodes
Tens of trapped ions feasible
Limitations in minimum trap size (~ 5µm)
Low temperature ( ~ -150C)
CNOT operation demonstrated
Trap region
37
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Alternative implementation techniques
Nuclear Magnetic Resonance: Qubits are the spin states of the nuclei of the
molecules of the liquid used (demonstrated up to 8 qubits)
Superconductors QC based on Josephson junctions: (~ 1K required), Charge
qubit/Flux qubit
Adiabatic Quantum Computer (D.Aharonov, W. Van Dam et al) based on Adiabatic
Theorem (simulated)
A Quantum System in its ground state remains in it along an adiabatic transformation
in which the Hamiltonian is varied slowly enough from an initial to a final one.
Idea: to vary the Hamiltonian slowly from initial to final state as if an U was performed
on the initial state. The final ground state encodes the solution
jiH 0
jjH ff
1,0,1 10 T
txHxHxxH
38
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Topological properties are deformation invariant (i.e. physically unaffected by
perturbations) : this would render quantum computation almost error-free
Topological Quantum computation
using non-abelian anyons(K. Shtengel, UC Riverside)
≡≠
C B A
Idea is to perform Quantum Computation using topological properties of quasi-particles
39
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Topological Quantum Computing
Topological differences between 2 and 3 dimensions Quantum systems (Leinaas & Myrheim, 1977)
)i.e. if two particles are confined in 2D, their trajectories involve non-trivial winding if their positions are interchanged twice(
2121 ,, rrerr i
Two identical particles exchange their position anticlockwise:
,0Boson, fermions
,0anyonsIn 2D the phase can take any value:
40
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Classes of trajectories taking N anyons from A to B are isomorphic to
BN
Non abelian anyons
A
B
Multiplication of elements of Bn is the
successive execution of the trajectories
Ni 11,
22 jijji
11111 Niiiiiii
12 i
2112
Topological Quantum computing –
41
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Topological Quantum Computing
Anyons might arise in some low dimensions
confined many particles systems
Quasiparticles = Localized disturbances of the quanto-
mechanical ground state of the two-dimensional
system
To check if quasiparticles are anyons:
1. Take quasiparticles around each other adiabatically (i.e. intial positions = final positions with interchange)
2. The adiabatic interchange applies a unitary transformation on the ground state (phase):
3. If anyons
dttRE dRRR R
0
Experimental evidence that quasiparticles occurring in fractional Quantum Hall
effect are (non abelian ) anyons is still debated (J. Goldman et al. 2006)
42
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Quantum Information – Topological techniques – Pairs of Anyons are brought together:
degeneracy is lifted -> 2 states = qubit
Qubitsin
U
U3 U1U2
Qubitsout
tMapping of unitary operations to braids not trivial
Readout of anyons not trivial
43
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Quantum Information – Conclusions and relevance to Particle physics
Quantum Computing promises breakthroughs in solving complex mathematical problems, some hard or insolvable classically (but still investigated)
Computational power is an obvious benefit for all scientific fields, including Particle Physics (e.g. searching through immense databases)
Simulation of Quantum Mechanical systems may be another area of research: essentially, to simulate a quantum mechanical system means really to simulate nature with its laws. This applies to
the world of nanotechnology as Particle Physics too.
Theoretically one could think of modified laws of Quantum Mechanics(e.g. ‘ad’ hoc’ terms,non linearities etc)
Use of Quantum technology for next generation of detectors
44
Rutherford Appleton LaboratoryParticle Physics Department
G. Villani QC seminar RAL 2008
Quantum Information – Backup slides -
Turing machine:Unbounded tape;Head that can read from the tape and can write on it, with infinite number of states;Instruction table.Given the initial head’s state and initial input the head reads, the table computes:The symbol the head writes on the tape;Where the head moves next on the tape.Church-Turing thesis: any effectively calculable function can be computed by a Turing machine
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