Quantum random walks – new method for designing quantum algorithms Andris Ambainis University of Latvia

Quantum computing New model of computing, based on quantum mechanics. More powerful than conventional (classical) computing.

Talk outline 1. Main results of quantum computing. 2. The model. 3. Quantum algorithms based on quantum walks.

Shor’s algorithm Factoring: given N=pq, find p and q. Best algorithm - 2 O(n 1/3 ), n – number of digits. Quantum algorithm - O(n 3 ) [Shor, 94]. Cryptosystems based on hardness of factoring/discrete log become insecure.

Grover's search Find i such that x i =1. Queries: ask i, get x i. Classically, N queries required. Quantum: O(  N) queries [Grover, 96]. Speeds up any search problem x1x1 x2x2 xNxN x3x3

NP-complete problems Does this graph have a Hamiltonian cycle? Hamiltonian cycles are: Easy to verify; Hard to find (too many possibilities).

Quantum algorithm Let N – number of possible Hamiltonian cycles. Black box = algorithm that verifies if the i th candidate – Hamiltonian cycle. Quantum algorithm with O(  N) steps x1x1 x2x2 xNxN x3x3 Applicable to any search problem

Pell’s equation Given d, find the smallest solution (x, y) to x 2 -dy 2 =1. Probably harder than factoring and discrete logarithm. Best classical algorithms: for factoring; 2 O(  N) for discrete logarithm. Hallgren, 2001: Quantum algorithm for Pell’s equation.

Number theory and algebraic problems Polynomial time quantum algorithms: Factoring [Shor, 94] Discrete logarithm [Shor, 94]; Pell’s equation [Hallgren, 02]. Principal ideal problem [Hallgren, 02]. Computing the unit group [Hallgren, 05].

Grover's search Find i such that x i =1. Queries: ask i, get x i. Classically, N queries required. Quantum: O(  N) queries [Grover, 96]. Speeds up any search problem x1x1 x2x2 xNxN x3x3

Amplitude amplification [Brassard et al., 01] Algorithm A that finds a certain object with probability . How many repetitions to achieve probability 2/3? Classically,  (1/  ). Quantum: O(1/  ). “Quantum black box”.

One way to design quantum algorithms Search procedure Amplify quantumly: Classical algorithm + amplitude amplification

Quantum counting Determine the fraction of x i =1. E.g., distinguish whether the fraction is  1/2-  or  1/2+ . Classical random sampling: O(1/  2 ) steps. Quantum: O(1/  ) steps x1x1 x2x2 xNxN x3x3 Can be used for more complicated statistics.

Element distinctness Numbers x 1, x 2,..., x N. Determine if two of them are equal. Classically: N queries. Quantum: O(N 2/3 ) x1x1 x2x2 xNxN x3x3

Triangle finding [Magniez, Santha, Szegedy, 03] Graph G with n vertices. n 2 variables x ij ; x ij =1 if there is an edge (i, j). Does G contain a triangle? Classically: O(n 2 ). Quantum: O(n 1.3 ).

The model of quantum computing

Probabilistic computation Probabilistic system with finite state space. Current state: probabilities p i to be in state i

Quantum computation Current state: amplitudes  i to be in state i i i 0.3 For most purposes, real (but negative) amplitudes suffice.

Probabilistic computation Pick the next state, depending on the current one /3 1/3

Probabilistic computation Transitions: r ij - probabilities to move from i to j. 2/3 1/3

Probabilistic computation  Probability vector (p 1, …, p M ).  Transitions: before the transition transition probabilities after the transition

Allowed transitions R –stochastic: If  i p i = 1, then  i p’ i = 1.

Quantum computation  Amplitude vector (  1, …,  M ),.  Transitions: before the transition transition matrix after the transition

Allowed transitions U – unitary: If, then.

Geometric interpretation (  1, …,  M ), - vectors on the unit sphere. Transformations that preserve - rotations of the unit sphere. 11 00

Summary so far Quantum  probabilistic with complex probabilities. Instead of  i p i = 1 we have (l 2 norm instead of l 1 ). How do we go from quantum world to conventional world?

Measurement Quantum state:   … +  M M |1|2|1|2 1 prob. |2|2|2|2 2 |M|2|M|2 M … Measurement

Quantum walks and quantum algorithms 1. Quantum random walks. 2. Grover’s quantum search algorithm. 3. Quantum walks + quantum algorithms.

Random walk on line Start at location 0. At each step, move left with probability ½, right with probability ½

Random walk on line State (x, d), x –location, d-direction. At each step, Let d=left with prob. ½, d=right w. prob. ½. (x, left) => (x-1, left); (x, right) => (x+1, right)

Quantum walk on line States |x, d , x –location, d-direction “Coin flip”: Shift:

Quantum walk on line Left: Right:

Quantum walk on line Left: Right:

Quantum walk on line Left: Right:

Quantum walk on line Left: Right:

Quantum walk on line Left: Right: 1

Quantum walk on line Left: Right: 1

Quantum walk on line Left: Right: 1

Quantum walk on line Left: Right: /2 1/8

Classical vs. quantum Run for t steps, measure the final location. Distance:  (  N) Distance:  (N)

Grover’s quantum search algorithm

Grover's search Find i such that x i =1. Queries: ask i, get x i x1x1 x2x2 xNxN x3x3

Queries in the quantum world States |1 , |2 , …, |N . Query: |i   |i , if x i =0; |i   -|i , if x i =1; (a state with amplitude 1 at location i gets mapped to a state with amplitude –1 at i)

Queries in the quantum world 123 … N … x i =1 12 … …

Grover’s algorithm times repeat: Query; Diffusion transformation:

Analysis of Grover’s algorithm For simplicity, assume exactly one i:x i =1. … … 1 st query

Diffusion transformation … Inversion against average … D

2 nd query … …

2 nd diffusion … …

Grover: summary Query+diffusion increases the amplitude of i:x i =1, decreases the amplitude of i:x i =0. Afterrepetitions the amplitude of i: x i =1 is almost 1. Measuring at this point gives the solution i. Why  N?

Probabilities vs. amplitudes Probabilities: 1/N probability to query the right i in 1 step. N steps to obtain the probability 1. Amplitudes: The amplitude of the right i is 1/  N. Each query+diffusion increases it by 2/  N. After  N steps it becomes 1. Measurement: amplitude 1  probability 1 2 =1.

Grid with N elements. Task: find the location storing a certain value. In one step, we can check the current location or move distance 1. Search on grids

[Benioff, 2000]  n*  n grid. Distance between opposite corners = 2  n. Grover’s algorithm takes steps. No quantum speedup. Quantum walks solve this problem!

Quantum walk search [Szegedy, 04] Finite search space. Some elements might be marked. Find a marked element! Perform a random walk, stop after finding a marked element.

Conditions on Markov chain Random walk must be symmetric: p xy =p yx. Start state = uniformly random state. T = expected time to reach marked state, if there is one

Main result [Szegedy, 04] Theorem Assume that: 1. There are no marked states, or 2. A marked state is reached in expected time at most T. A quantum algorithm can distinguish the two cases in time O(  T). Quadratic speedup for a variety of problems.

Application 1 N states. Is there a marked state? Random walk: at each step move to a randomly chosen vertex. Finds a marked vertex in N expected steps. Quantum: O(  N) steps [Grover]

Application 2: search on grids Random walk: at each step move to a random neighbour. Finding marked state: O(N log N) steps. Quantum algorithm: [A, Kempe, Rivosh, 2005]

Application 3: element distinctness Numbers x 1, x 2,..., x N. Determine if two of them are equal. Well studied problem in classical CS. Classically: N steps x1x1 x2x2 xNxN x3x3

Johnson graph Vertices: sets S, |S|=k, S  {1, 2, …, N}. Edges: (S, T), for S, T that differ only in 1 element. {1, 2, 3} {1, 2, 4}{1, 3, 5} {2, 3, 4}{3, 4, 5}

Random walk Marked vertices = those for which S contains i, j: x i = x j. Random walk in which we maintain x i, i  S. {1, 2, 3} {1, 2, 4}{1, 3, 5} {2, 3, 4}{3, 4, 5} K queries to start; 1 query to move to adjacent vertex.

Search by random walk Time to find a marked vertex: K queries to start; moving steps. {1, 2, 3} {1, 2, 4}{1, 3, 5} {2, 3, 4}{3, 4, 5} Quantum:

Quantum algorithm Quantum time: K queries to start; moving steps. Total: {1, 2, 3} {1, 2, 4}{1, 3, 5} {2, 3, 4}{3, 4, 5} K=N 2/3 O(N 2/3 )

Triangle finding [Magniez, Santha, Szegedy, 03] Graph G with n vertices. n 2 variables x ij ; x ij =1 if there is an edge (i, j). Does G contain a triangle? Classically: O(n 2 ). Quantum: O(n 1.3 ).

Matrix multiplication [Buhrman, Špalek, 05] A, B, C – n*n matrices. Finding C=AB: O(n 2.37… ) steps; Given A, B and C, we can test AB=C in: O(n 2 ) steps by a probabilistic algorithm; O(n 5/3 ) steps by a quantum algorithm.

Inside the Szegedy’s black box How do we turn random walks into quantum algorithms?

Quantum walk State space with states |i . Starting state Quantum walk with two transition rules: “usual” for unmarked vertices; “special” for marked

Quantum walk algorithm No marked vertices: The starting state is unchanged. Some marked vertices: The starting state is changed at the marked vertices. Changes spread and accumulate. Run for sufficiently many steps, test if the system is still in the starting state.

As in Grover’s search… ……

Mathematics - vector of probabilities 1 step: P’ = M  P k steps: P’ = M k  P Eigenvectors and eigenvalues of M, M – transition matrix.

Mathematics II Eigenvectors of M: v 1, …, v N. M v i = i v i M t v i = ( i ) t v i Express starting distribution via eigenvectors:

Mathematics III Random walk Probability matrix M; Expected time T to hit a marked vertex. Quantum walk Unitary matrix U; Time T’ at which the state becomes sufficiently different from the starting state.

Other applications of quantum walks

“Glued trees” [Childs et al., 02] Two full binary trees of depth d; Randomly connect leaves of the two graphs.

“Glued trees” [Childs et al., 02] Start position: the left root. Task: find the other root. Vertices and edges not labeled. 2 d+2 -2 vertices. Any classical algorithm takes  (c d ) steps.

“Glued trees” [Childs et al., 02] Perform a quantum walk, starting from the left root. After O(d 2 ) steps, a quantum walk is at the right root. Exponential speedup: O(d 2 ) vs  (c d ).

[A, Childs, et al., 07] AND-OR formula of size M. Variables accessed by queries: ask i, get x i. Theorem Any formula can be evaluated with O(M 1/2+o(1) ) queries. ANDOR 1x11x1 2x22x2 3x33x3 4x44x4 Speedup for anything that can be expressed as a formula

Conclusion Quantum walks can be used to design quantum algorithms for many problems. Quantum walk search: create a classical random walk, quantize it with a speedup. “Quantum black box” within a classical algorithm.

Building a quantum computer Classical computer operates on bits (0 or 1). Quantum bit: a system with basis states |0 , |1 , general state  0 |0  +  1 |1 . Need physical systems that act like quantum bits… 10s of candidates…

NMR quantum computing (IBM, Waterloo, etc.) Quantum computer = molecule. Quantum bits = nuclear spins. Operations performed using magnetic fields. Spin – property that determines how the particle behave in a magnetic field  0,  1

NMR quantum computing 12 quantum bits (IQC): Creating a certain quantum state. 7 quantum bits (IBM): Factoring 15; Grover’s search among 4 items.

Quantum cryptography Secure data transmission, using quantum mechanics. Only requires single quantum bits. Implemented over 200km distance. Available commercially. QKD device of Id Quantique (Geneva)