Introduction to Quantum Computation Neil Shenvi Department of Chemistry Yale University

Talk Outline Background What is Quantum Computation? Quantum Algorithms Decoherence and Noise Implementations Applications Quantum Random Walks O Noise in Grover’s Algorithm Decoherence in Spin Systems

Background: Classical Computation C:\Hello.exe Hello World! InputComputationOutput What is the essence of computation?

Classical Computation Theory Church-Turing Thesis: Computation is anything that can be done by a Turing machine. This definition coincides with our intuitive ideas of computation: addition, multiplication, binary logic, etc… What is a Turing machine? … … Infinite tape Read/Write head Finite State Automaton (control module) … … Computation … … Output … … Input

Classical Complexity P NP NP-complete Some problems are more difficult than others. Polynomial hierarchy Require polynomial time to solve Require exponential time to solve All Turing machine-equivalent computers have an identical hierarchy. Require exponential(?) time to solve

Classical Complexity P NP NP-complete Some important problems do not have known classical polynomial algorithm and or a known place in the hierarchy. Polynomial hierarchy Factoring Graph Isomorphism ? ? Best known algorithm to factor N-digit number: Time ~ Exp(N 1/3 ) Best known algorithm to compare two N-node graphs: Time ~ Exp(N)

Classical Computation Theory What kind of systems can perform universal computation? Desktop computers Billiard balls DNA Cellular automata These can all be shown to be equivalent to each other and to a Turing machine! The Big Question: What next?

Talk Outline Background What is Quantum Computation? Quantum Algorithms Decoherence and Noise Implementations Applications

What Is Quantum Computation? Conventional computers, no matter how exotic, all obey the laws of classical physics. On the other hand, a quantum computer obeys the laws of quantum physics.

The Bit The basic component of a classical computer is the bit, a single binary variable of value 0 or The state of a classical computer is described by some long bit string of 0s and 1s At any given time, the value of a bit is either ‘0’ or ‘1’.

The Qubit A quantum bit, or qubit, is a two-state system which obeys the laws of quantum mechanics. =|1  =|0  Valid qubit states: |  = |0  |  = |1  |  = (|0  - e i  /4 |1  )/  2 |  = (2|0  - 3e i5  /6 |1  )/  13 Spin-½ particle The state of a qubit |  can be thought of as a vector in a two-dimensional Hilbert Space, H 2, spanned by the Basis vectors |0  and |1 .

Computation with Qubits How does the use of qubits affect computation? Classical Computation Data unit: bit x = 0 x = Valid states: x = ‘0’ or ‘1’ |  = c 1 |0  + c 2 |1  Quantum Computation Data unit: qubit Valid states: |  = |0  |  = |1  |  = (|0  + |1  )/√2 =|1  =|0  = ‘1’ = ‘0’

Computation with Qubits How does the use of qubits affect computation? Classical Computation Operations: logical Valid operations: AND = 0i -i NOT = in out in bit 2-bit Quantum Computation Operations: unitary Valid operations: σ X = σ y = σ z = H d = CNOT = √2 1 1-qubit 2-qubit

Computation with Qubits How does the use of qubits affect computation? Classical Computation Measurement: deterministic x = ‘0’ State Result of measurement ‘0’ x = ‘1’ ‘1’ Quantum Computation Measurement: stochastic |  = |0  |  = |0  - |1  State Result of measurement |  = |1  22 ‘0’ ‘1’ ‘0’ 50% ‘1’ 50%

More than one qubit u 11 u 12 u 21 u 22 Single qubit c1c1 c2c2 c1c1 c2c2 Two qubits H 2 = , |0,|1|0,|1 H 2  2 = H 2  H 2 =, |00 ,|01 ,|10 ,|11  , , c1c1 c2c2 c3c3 c4c4 c1c1 c2c2 c3c3 c4c4 u 11 u 12 u 13 u 14 u 21 u 22 u 23 u 24 u 31 u 32 u 33 u 34 u 41 u 42 u 43 u 44 Hilbert space U|  = U|  = Operator |  = c 1 |0  + c 2 |1  = |  c 1 |00  + c 2 |01  + c 3 |10  + c 4 |11  = = Arbitrary state

Quantum Circuit Model σ x  I = CNOT = |0|0 |0|0 |1|1 |0|0 |1|1 |1|1 ‘1’ Example Circuit σxσx One-qubit operation CNOT Two-qubit operation Measurement

Quantum Circuit Model 1/√ σxσx CNOT |0  + |1  |0|0 Example Circuit √2 ______ 1/√ |0  + |1  |0|0 √2 ______ ‘0’ or ‘1’ or 50% Separable state: can be written as tensor product |  = |   |  Entangled state: cannot be written as tensor product |  ≠ |   |  ? ?

Some Interesting Consequences Quantum Superordinacy All classical quantum computations can be performed by a quantum computer. U No cloning theorem It is impossible to exactly copy an unknown quantum state |  |0|0 Reversibility Since quantum mechanics is reversible (dynamics are unitary), quantum computation is reversible. |  |  | 

Talk Outline Background What is Quantum Computation? Quantum Algorithms Decoherence and Noise Implementations Applications

Quantum Algorithms: What can quantum computers do? Grover’s search algorithm Quantum random walk search algorithm Shor’s Factoring Algorithm

Grover’s Search Algorithm Imagine we are looking for the solution to a problem with N possible solutions. We have a black box (or ``oracle”) that can check whether a given answer is correct. 78 Question: I’m thinking of a number between 1 and 100. What is it? Oracle No 3 Oracle Yes

Grover’s Search Algorithm The best a classical computer can do on average is N/2 queries. 1 Oracle No... 2 Oracle No 3 Oracle Yes Classical computer Oracle No+No+Yes+No+... Quantum computer Using Grover’s algorithm, a quantum computer can find the answer in  N queries! Superposition over all N possible inputs.

Grover’s Search Algorithm Pros: Can be used on any unstructured search problem, even NP-complete problems. Cons: Only a quadratic speed-up over classical search. The circuit is not complicated, but it doesn’t provide an immediately intuitive picture of how the algorithm works. Are there any more intuitive models for quantum search? O σzσz O σzσz … … … … |0|0 |0|0 |0|0 O(  N) iterations HdHd HdHd HdHd … HdHd HdHd HdHd … HdHd HdHd HdHd … HdHd HdHd HdHd … HdHd HdHd HdHd

Quantum Random Walk Search Algorithm Idea: extend classical random walk formalism to quantum mechanics Classical random walk:

Quantum Random Walk Search Algorithm To obtain a search algorithm, we use our “black box” to apply a different type of coin operator, C 1, at the marked node C0C0 C1C C0=C0= 1 2 C1=C1=

Quantum Random Walk Search Algorithm Pros: As general as Grover’s search algorithm. Cons: Same complexity as Grover’s search algorithm. Slightly more complicated in implementation Slightly more memory used Interesting Feature: Search algorithm flows naturally out of random walk formalism. Motivation for new QRW- based algorithms?

Shor’s Factoring Algorithm Find the factors of: 57 3 x 19 Find the factors of: whimper All known algorithms for factoring an n-bit number on a classical computer take time proportional to O(n!). But Shor’s algorithm for factoring on a quantum computer takes time proportional to O(n 2 log n). Makes use of quantum Fourier Transform, which is exponentially faster than classical FFT.

# bits factoring in years5x10 15 years3x10 29 years factoring in years10 12 years7x10 25 years factoring in days3x10 8 years2x10 22 years with a classical computer # bits # qubits # gates3x10 9 2X10 11 X10 12 factoring time4.5 min36 min4.8 hours with potential quantum computer (e.g., clock speed 100 MHz) R. J. Hughes, LA-UR Shor’s Factoring Algorithm The details of Shor’s factoring algorithm are more complicated than Grover’s search algorithm, but the results are clear:

Talk Outline Background What is Quantum Computation? Quantum Algorithms Decoherence and Noise Implementations Applications

Decoherence and Noise What happens to a qubit when it interacts with an environment? Quantum computer Environment V Quantum information is lost through decoherence. σ1σ1 σ2σ2 σ3σ3 σNσN …

Types of Decoherence T 1 processes: longitudinal relaxation, energy is lost to the environment V T 2 processes: transverse relaxation, system becomes entangled with the environment V + + What are the effects of decoherence?

Effects of Environment on Quantum Memory Fidelity of stored information decays with time. T 1 – timescale of longitudinal relaxation T 2 – timescale of transverse relaxation

Effects of Environment on Quantum Algorithms Errors accumulate, lowering success rate of algorithm Grover’s algorithm success rate n = # of qubits O O Ideal oracle Noisy oracle

Suppressing Decoherence 1. Remove or reduce V, i.e. build a better computer System isolated from environment 2. Increase B, i.e. increase level splitting B E |0|0 |1|1 When  E >> V, decoherence is small EE 3.Use decoherence free subspace (DFS) 4. Use pulse sequence to remove decoherence

Talk Outline Background What is Quantum Computation? Quantum Algorithms Decoherence and Noise Implementations Applications

Some Proposed Implementations for QC NMR B Ion trap Optical Lattice Kane Proposal

The Loss-Divincenzo Proposal D. Loss and D.P. DiVincenzo, Phys. Rev. A 57, 120 (1998); G. Burkhard, H.A. Engel, and D. Loss, Fortschr. der Physik 48, 965 (2000).

Solid State Electron Spin Qubit Silicon lattice Phosphorus impurity Electron wavefunction Si 28 (no spin) Si 29 (spin ½) External Magnetic Field, B Hyperfine coupling Dipolar coupling

System Hamiltonian Electron spin N nuclear spins Hyperfine couplingDipolar coupling ~10 5 Hz~10 2 Hz ~10 7 Hz / T ~10 11 Hz / T

Hyperfine-Induced Longitudinal Decay For B > B c, T 1 is infinite Critical field for electron spin relaxation:

Hyperfine-Induced Transverse Decay Free evolution Spin echo pulse sequence Spin echo pulse sequence removes nearly all dephasing!

Talk Outline Background What is Quantum Computation? Quantum Algorithms Decoherence and Noise Implementations Applications

Factoring – RSA encryption Quantum simulation Spin-off technology – spintronics, quantum cryptography Spin-off theory – complexity theory, DMRG theory, N-representability theory

Acknowledgements Dr. Julia Kempe, Dr. Ken Brown, Sabrina Leslie, Dr. Rogerio de Sousa Dr. K. Birgitta Whaley Dr. Christina Shenvi Dr. John Tully and the Tully Group