Algorithms Artur Ekert

Our golden sequence H H

Circuit complexity n QUBITS B A A B B B B A # of gates (n) = size of the circuit (n) # of parallel units (n) = depth of the circuit (n) n qubit circuit operation described by 2 n x 2 n unitary matrix size and depth of circuits grow with n Fix your building units, a finite set of adequate gates A, B, C… Size 12 Depth 4

Asymptotic notation for comparisons

Asymptotic notation

Asymptotic notation - example is both and hence it is

Quantifying growth exponential quadratic or polynomial linear root-n logarithmic cubic or polynomial

Efficient quantum algorithms B B A A B B A B B A A B A

Quantum Hadamard Transform H H H H H H H H

Discrete set of phase gates Control phase gates = Insidious phases…

Quantum Fourier Transform H H H H H H F1F1 F2F2 F3F3

Is also known as the quantum Fourier transform on group group = the set with operation (addition mod 2) group = the set with operation (addition mod 2 bit by bit) example for n=15 Aside – Hadamard is Fourier

Quantum Fourier transform on group Aside – Hadamard is Fourier

Quantum function evaluation f Boolean function

Quantum function evaluation can be viewed as m Boolean functions f m-1 f m-2 f0f0 …………………………………………

Query Scenario f very precious, you are charged fixed amount of money each time you use it BLACK BOX, ORACLE INPUT: is a function f given as an ORACLE GOAL: is to determine some properties of f making as few queries to f as possible

Quantum interferometry revisited H H H H U

Phases in a new way H H U

Deutsch’s Problem f David Deutsch Given CONSTANT BALANCED ? is f constant or balanced four possible oracles

Deutsch’s Problem H H f f f CONSTANT BALANCED 2 queries + 1 auxiliary operation 1 query + 2 auxiliary operations Quantum Classical

Deutsch’s Problem – The Guts H H f H

H H f H But…

Deutsch’s Problem – The Guts H H f H So, it is now clear what happens if f(0) and f(1) are the same or different….

Deutsch’s Problem Generalised H H f CONSTANT any other output BALANCED H H H H H H H H INPUT: either constant or balanced PROMISE: OUTPUT: determine whether constant or balanced CLASSICAL COMPLEXITY: queries

Deutsch’s Problem Generalised HH f

HH f

HH f

What is the amplitude for finding the register in the |0> state? If f(x) constant, this has amplitude 1 i.e. it is guaranteed If f(x) balanced, this has amplitude 0 i.e. it will never happen

Deutsch’s Problem Generalised H H f CONSTANT any other output BALANCED H H H H H H H H INPUT: either constant or balanced PROMISE: OUTPUT: determine whether constant or balanced CLASSICAL COMPLEXITY: queries

Fair comparison? classical deterministic: quantum : 1 classical probabilistic with error prob. : Query in k places, if the queries had at least one 0 and one 1 then the function is balanced, otherwise assume it is constant. Probability that it is balanced when declared constant is FAIR COMPARISON

Bernstein-Vazirani Problem H H f H H H H H H H H INPUT: is of the form PROMISE: OUTPUT: binary string

Search Problem INPUT: PROMISE: OUTPUT: binary string Searching large and unsorted database containing 2 n items Example of a sorted database: a phone book if you are given a name and looking for a telephone number n lookups suffice Example of an unsorted database: a phone book if you are given a number and looking for a name you need to check 2 n items before you succeed with probability P=1 you need to check 2 n-1 items before you succeed with probability P=0.5 Classical Complexity:

Grover’s algorithm It is easy to recognize a solution, although hard to find it.

Grover’s algorithm H H f H H H H H H INPUT: PROMISE: OUTPUT: binary string f0f0 H H H H H f H H H f0f0 H H H H ITERATION 1 ITERATION 2 ………………………………………………………… Quantum Complexity:

Grover’s algorithm H f H H H f0f0 H H H H ITERATION

Grover’s algorithm H f H H H f0f0 H H H H ITERATION

Grover’s algorithm H f H H H f0f0 H H H H

H H H H H H H H is the state input at the start of the iterations

Grover’s algorithm Geometric Interpretation: Reflects a state about the axis orthogonal to So, we need to consider the composed, repeated actions of and

Grover’s algorithm sin  = | |

Grover’s algorithm Overall action: Rotation by angle 2 

Grover’s algorithm H H H H H H H H

H H f H H H H H H f0f0 H H H H H f H H H f0f0 H H H H ITERATION 1 ITERATION 2 …………………………………………………………

Grover’s algorithm After r iterations, the state is rotated by from the hyperplane for large n We iterate until

Query complexity quantum : classical probabilistic: Quadratic speedup compared to classical search algorithms Cryptanalysis: Attack on classical cryptographic schemes such as DES (the Data Encryption Standard) essentially requires a search among 2 56 =7 £ possible keys. If these can be checked at a rate of, say, one million keys per second, a classical computer would need over a thousand years to discover the correct key while a quantum computer using Grover's algorithm would do it in less than four minutes.

Applications of Grover Most common example is an unsorted database. Not a common scenario! Finding most efficient route between two places on a map. Brute-force code breaking (such as the DES example we’ve just seen). Any classical algorithm with probabilistic outcome can be enhanced.

Simon’s Problem INPUT: PROMISE: OUTPUT: period Example: s=110 is the period (in the group) Classical Complexity:

Fields and vector spaces over them

Binary vectors binary vectors Inner product

Binary vectors vectors x a binary vector can be perpendicular to itself

Binary vectors

Quantum Hadamard Transform H H H H H H H H

Simon’s algorithm n qubits HH y f(x)

Simon’s algorithm n qubits HH

Simon’s algorithm n qubits HH Solve the system of linear equations Probability of failure of generating linearly independent vectors y is less than 0.75 Needs roughly n queries. Quantum complexity

Classical Complexity Analysis Classical approach: Randomly choose: Evaluate: Search for collisions: Average number of collisions: Number of queries in a classical probabilistic approach : CLASSICAL Probability of at least one collision:

Quantum Complexity Analysis 1

Quantum Complexity Analysis 2

Summary HH f Deutsch (1985), Deutsch and Jozsa (92): The first indication that quantum computers can perform better H f Grover: Polynomial separation Simon: Exponential separation HH f H f0f0 f0f0 HH f classical quantum classical quantum

More general oracles Quantum oracles do not have to be of this form n qubits m qubits e.g. generalized controlled-U operation

Phase estimation problem n qubits m qubits

Phase estimation algorithm n qubits m qubits H STEP 1: Recall Quantum Fourier Transform:

Phase estimation algorithm n qubits m qubits H STEP 2: Apply the reverse of the Quantum Fourier Transform FyFy But what if p has more than n bits in its binary representation ?

Arbitrary phases

geometric series! Final state Probability of result

Phase estimation algorithm Probability

Phase estimation - solution n qubits m qubits H FyFy

Group This is a group under multiplication mod M For example

Order-finding problem For example

Order finding problem Input: M (an m -bit integer) and a  {1,2,…, M  1} such that gcd( a, M ) = 1 } Output: ord M ( a ), which is the minimum r > 0 such that a r = 1 ( mod M ) No efficient classical algorithm is known for this problem

Order finding problem period 6

Solving order-finding via phase estimation n qubits m qubits Suppose we can build a circuit that multiplies y by the powers of a, but what about the eigenvector u ? n qubits m qubits

Sum over all possible values of k - the same Sum, different order of terms Eigenvalues and eigenvectors of U It seems that we need to know r - do we ?

F FyFy Estimate of p 1 with prob. |  | 2 Estimate of p 2 with prob. |  | 2 Two or more eigenvectors…

Solving order-finding via phase estimation

F FyFy Randomly chosen q Co-prime with r ? Continuous fractions 2m qubits m qubits We need to distinguish between 1/r and 1/(r+1) and r is of the order of M which is an m bit number Solving order-finding via phase estimation

The integer factorization problem Input: M ( m -bit integer; we can assume it is composite) Output: p, q (each greater than 1 ) such that pq = M No efficient (polynomial-time) classical algorithm is known for this problem. Important for public key cryptosystems Order finding and factoring have the same complexity. Any efficient algorithm for one is convertible into an efficient algorithm for the other.

Math behind quantum factoring latter event occurs with probability  ½

Truth by example…

Proposed quantum algorithm (repeatedly do): 1.randomly choose a  {2, 3, …, M  1} 2.compute g = gcd (a,M ) 3. if g > 1 then output g, M/g else compute r = ord M ( a ) quantum part ! if r is even then compute x = a r /2 +1 mod M compute h = gcd (x,M ) if h > 1 then output h, M /h Quantum factoring

Hidden subgroup problem Given constant and distinct on cosets of subgroub K Find a generating set for K in polylog |G| steps

INPUT: PROMISE: OUTPUT: period Example: s=110 is the period (in the group) Classical Complexity: Simon’s algorithm revisited

n qubits HH

Simon’s Problem as HSP

Simon’s algorithm: G = ( Z 2 ) n K = {0, r } Shor’s algorithm: G = Z and K = r Z symmetric group graph isomorphism dihedral group shortest vector in a lattice D8 D8 ABCD S4S4 Examples OPEN PROBLEMS QC NEEDS YOU

Find the shortest vector Open problems Graph isomorphism