Introduction to Quantum Computation Neil Shenvi Department of

Introduction to Quantum Computation Neil Shenvi Department of Chemistry Yale University

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

Background: Classical Computation Input Computation Output 2 + 2 C: Hello. exe 4 Hello World! 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? … 0100101101010010110… Input Finite State Automaton (control module) Infinite tape Computation … 000000101111100… Read/Write head … 0100101101010010110… … 1110010110100111101… Output

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

Talk Outline l Background l What is Quantum Computation? l Quantum Algorithms l Decoherence and Noise l Implementations l 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 1. 1 0 0 1 At any given time, the value of a bit is either ‘ 0’ or ‘ 1’. The state of a classical computer is described by some long bit string of 0 s and 1 s. 00010101101101010011010110. . .

The Qubit A quantum bit, or qubit, is a two-state system which obeys the laws of quantum mechanics. Valid qubit states: Spin-½ particle =|1 =|0 | = |0 | = |1 | = (|0 - ei /4 |1 )/ 2 | = (2|0 - 3 ei 5 /6 |1 )/ 13 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: qubit Data unit: bit = ‘ 1’ = ‘ 0’ =|1 =|0 Valid states: | = c 1|0 + c 2|1 x = ‘ 0’ or ‘ 1’ x=1 x=0 Quantum Computation | = |0 0 0 1 1 | = |1 | = (|0 + |1 )/√ 2

Computation with Qubits How does the use of qubits affect computation? Classical Computation Quantum Computation Operations: unitary Valid operations: Operations: logical Valid operations: in NOT = σX = 0 1 1 0 out 1 -bit 1 -qubit σy = 0 1 1 0 0 i -i 0 in in 0 1 0 0 0 1 2 -bit out 2 -qubit CNOT = 1 0 0 -1 Hd = 1 √ 2 1 0 0 1 AND = σz = 0 1 0 0 0 1 0 1 1 1 -1

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

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

Quantum Circuit Model Example Circuit Two-qubit operation One-qubit operation |0 |1 σx |0 1 0 0 0 1 0 σx I = 0 0 1 0 01 00 00 10 0 1 0 0 Measurement CNOT = 1 0 0 |1 ‘ 1’ 0 0 0 1 0

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

Some Interesting Consequences Reversibility Since quantum mechanics is reversible (dynamics are unitary), quantum computation is reversible. |0000 | |0000 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 | |

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

Quantum Algorithms: What can quantum computers do? l Grover’s search algorithm l Quantum random walk search algorithm l 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. Question: I’m thinking of a number between 1 and 100. What is it? 78 Oracle No 3 Oracle Yes

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

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. O( N) iterations Hd … Hd O H d σz H d Hd Hd … |0 |0 Hd Hd … … … 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?

Quantum Random Walk Search Algorithm Idea: extend classical random walk formalism to quantum mechanics Classical random walk: Quantum random walk: Moves walkers based on coin Flips coin

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 C 1 C 0= 1 2 1 -1 -1 1 -1 -1 1 C 1= -1 0 0 0 0 -1

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 QRWbased algorithms?

Shor’s Factoring Algorithm Makes use of quantum Fourier Transform, which is exponentially faster than classical FFT. Find the factors of: 57 Find the factors of: 16238476016501762387610762691722612171239872103974621876187 12073623846129873982634897121861102379691863198276319276121 3 x 19 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).

Shor’s Factoring Algorithm The details of Shor’s factoring algorithm are more complicated than Grover’s search algorithm, but the results are clear: with a classical computer # bits factoring in 2006 factoring in 2024 factoring in 2042 1024 105 years 38 years 3 days 2048 5 x 1015 years 1012 years 3 x 108 years 4096 3 x 1029 years 7 x 1025 years 2 x 1022 years with potential quantum computer (e. g. , clock speed 100 MHz) # bits # qubits # gates factoring time 1024 5124 3 x 109 4. 5 min 2048 10244 2 X 1011 36 min 4096 20484 X 1012 4. 8 hours R. J. Hughes, LA-UR-97 -4986

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

Decoherence and Noise What happens to a qubit when it interacts with an environment? Environment Quantum computer V σ1 σ2 σ3 … Quantum information is lost through decoherence. σN

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 T 1 – timescale of longitudinal relaxation T 2 – timescale of transverse relaxation Fidelity of stored information decays with time.

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

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 |1 E When E >> V, decoherence is small E |0 B 3. Use decoherence free subspace (DFS) 4. Use pulse sequence to remove decoherence

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

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

The Loss-Divincenzo Proposal D. Loss and D. P. Di. Vincenzo, 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 Electron wavefunction Si 28 (no spin) Phosphorus impurity Dipolar coupling Si 29 (spin ½) Hyperfine coupling Silicon lattice External Magnetic Field, B

System Hamiltonian Hyperfine coupling Electron spin ~1011 Hz / T Dipolar coupling N nuclear spins ~107 Hz / T ~105 Hz ~102 Hz

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

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

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

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

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