# Notes on Quantum Computation

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Notes on Quantum Computation
E.G. Villani STFC Rutherford Appleton Laboratory

Outline Quantum Computation introduction Algorithms examples
Quantum Computation Technology Conclusions

Introduction notes Topics: Quantum information and computation theory Implementation using different technologies Topological Quantum computing Quantum magnetism … 25th Winter School in Theoretical Physics Institute for Advanced Studies Jerusalem 2007

Jerusalem notes Unified (east and west) 1967
Moslem Christian Jewish Armenian Unified (east and west) 1967 Capital city of Israel from 1980 Around 800,00 inhabitants : approx 70% Jewish, 29% Muslim, 1% Christian Mostly religious: ‘secular’ population in constant decline Old city around 1km2 :one of the oldest cities in the world

Jerusalem notes Dome of the ‘Rock’ 7th century AC Western Wall
Retaining wall for the 2nd Temple , 1st century BC Holy Sepulchre Initially built by Constantine 4th Century BC

Computation –introduction -
The general process of computation can be described as an operation performed on initial information and the reading out of the results: Input Output Op Computation can be performed in classical domain using analogue or digital blocks

Classical Computation – analogue -
Analogue computation example: PID controller

Analogue Computation – analogue -
Analogue Sallen Key 5th order low pass filter

Classical computation – digital -
Digital Sallen Key 5th order low pass filter IIR implementation

Quantum limit Moore’s Law (1965): Exponential size shrinking of electronic devices -> atomic limit will be approached around 2015

Quantum Computing –introduction -
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

Quantum Computing –introduction -
computation in the Quantum domain A composite system of N Quantum bits (Qubits) is the input operator input output An unitary operator is applied A non unitary operator is applied to perform measurement

Quantum Information –introduction -
Quantum information: qubits A Quantum bit (qubit) describes the states of each individual two-level systems. In the computational bases: 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.

Quantum Information In a Quantum computer a unit vector in a Hilbert space H describes the initial state of the system An input state vector of non interacting qubits can be written in the form: A state vector of interacting qubits consists of entangled states E.g.:

Quantum Information In a more general case, a state of n qubits can be found in a mixed state: The density operator used to describe the state of a subsystem, by tracing out the unwanted system

Quantum computing –introduction -
Quantum processing: Evolution To perform a Quantum computation an unitary operation is performed on the qubits system: The generator Hamiltonian H has to generate this evolution according to Schrodingers’ equation: The Hamiltonian has to be found for a specific operation U:

a classical computer does)
Quantum computing If U unitary a solution for H always exists. 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. 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)

Quantum computing Quantum result: Measurement
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 computing – general process -
General Quantum circuit model U Superoperator: Quantum operation acts on an input density operator plus ancillary register If unchanged dimension of H input op output

Quantum computing Quantum gates
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: A Quantum gate U rotates the Bloch vector on the Bloch sphere

Quantum computing Universal Quantum gates
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 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 Efficiency of approximation ( number of gates) using G1(includes inverses)

U H X Quantum computing Quantum measurement
V. Neumann measurement with respect to any orthonormal basis V. Neumann measurement U H X Example: Computational basis to Bell basis

Bell X Z Quantum computing Quantum communication - teleportation a
Using two classical bits , it is possible to send the state of a qubit: quantum state transmitted using classical channels! a Alice Bell b X Z Bob Quantum teleportation useful to implement 2-Qubit gates First Quantum teleportation experimentally achieved using photons 1998

Quantum computing – algorithms -
First Quantum algorithm- Deutsch Jozsa Oracle U Σ Quantum superposition and interference

Quantum computing – algorithms -
Define the reversible mapping: Deutsch Quantum algorithm Problem: To determine if f is constant or balanced. Classically: 2 queries Deutsch algorithm: 1 query Input an eigenstate to the target qubit of an operator and associate the eigenvalue with the control register

Quantum computing – algorithms -
Deutsch Quantum algorithm H H Simultaneous computation result

Quantum computing – algorithms -
Deutsch Jozsa Quantum algorithm H H Problem: H H H H To determine if f is constant or balanced. Classically: 2n-1+1queries Deutsch algorithm: 1 query Exponential increase in efficiency

Quantum computing – algorithms -
Quantum algorithm – Quantum Fourier Transform Problem: 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 Shor’s algorithm for N factoring could compute 100s digits in seconds Order of random element in ZN Quantum complexity QFT QFT-1 Classical complexity Factorization believed to be NP problem but not demonstrated

Quantum computing – algorithms -
R(ivest)S(hamir)A(dleman) cryptosystem N,E Alice Bob P,Q : N = PQ M E: GCD(E,(P-1)(Q-1)=1 MEmod N E-1mod (P-1)(Q-1) (ME)E-1mod N =M Difficult to factor large numbers: classically ~ weeks for 100 digits N: doubling the digits implies a factorization ~ 106 years!

Quantum computing – algorithms -
Quantum algorithms remarks 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 op input output Acyclic quantum gate arrays can compute in polynomial time any function computable in polynomial time by CTM.

Quantum computing – further algorithms -
Any P problem can be mapped onto a acyclic quantum gate Cyclical quantum gate still not investigated: Compactness Phase delay 1-qubit cyclic gate represented by U(2) group 2-qubit cyclic gate represented by U(2) group

Quantum computing – further algorithms -
Perturbation of a 2-qubit cyclic network via C-NOT gate Evolution of a cyclic network after n cycles: In U eigenbases: Simplified QFT Simplified form for Quantum Gates (i.e. FIR vs IIR in DF) Phase delay may lead to instability Quantum oscillator!

Quantum Computing – implementation techniques
Technical issues for building a Quantum Computer (DiVincenzo 2000) 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/ faster than decoherence time Reliable representation of Quantum Information (scalable number of Qubits ) Setting of initial state of qubits Quantum gates reliable (decoherence) Readout Several proposed solution for quantum computation Alternative proposed solution to decoherence and error problem is topological quantum computation

Implementation techniques – Quantum Dots
InAs/GaAs SA QD grown with MBE Quantum dots using lithography : 100nm spacing Good reproducibility Nanotechnology required Contamination 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 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 T=5K in GaAs)

Implementation techniques – Quantum Dots
AC SEM of double Q-dot device 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… RO using Spin to Q technique

Implementation techniques – Ion traps -
Trap region 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 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

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

Topological Quantum computation using non-abelian anyons (K
Topological Quantum computation using non-abelian anyons (K. Shtengel, UC Riverside) Topological properties are deformation invariant (i.e. physically unaffected by perturbations) : this would render quantum computation almost error-free C B A Idea is to perform Quantum Computation using topological properties of quasi-particles

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) Two identical particles exchange their position anticlockwise: Boson, fermions In 2D the phase can take any value: anyons

Topological Quantum computing –
Classes of trajectories taking N anyons from A to B are isomorphic to BN B A Multiplication of elements of Bn is the successive execution of the trajectories Non abelian anyons

Topological Quantum Computing
Anyons might arise in some low dimensions confined many particles systems To check if quasiparticles are anyons: Take quasiparticles around each other adiabatically (i.e. intial positions = final positions with interchange) The adiabatic interchange applies a unitary transformation on the ground state (phase): If anyons Quasiparticles = Localized disturbances of the quanto-mechanical ground state of the two-dimensional system Experimental evidence that quasiparticles occurring in fractional Quantum Hall effect are (non abelian ) anyons is still debated (J. Goldman et al. 2006)

Quantum Information – Topological techniques –
Pairs of Anyons are brought together: degeneracy is lifted -> 2 states = qubit Mapping of unitary operations to braids not trivial t Readout of anyons not trivial U3 U2 U1 Qubits out Qubits in U

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

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