Exact quantum algorithms Andris Ambainis University of Latvia

Types of quantum algorithms  Bounded-error: correct answer with probability at least 2/3.  Exact: correct answer with certainty (probability 1).

Grover's search  Is there i:x i =1?  Classically, N queries required.  Quantum: O(  N) queries [Grover, 96].  Quantum, exact: N queries x1x1 x2x2 xNxN x3x3

Model

Query model  Function f(x 1,..., x N ), x i  {0,1}.  x i given by a black box: i xixi Complexity = number of queries

Queries in the quantum world  Basis states: |1,1 , |1, 2 , …, |N, M .  Query: |i, j   |i, j , if x i =0; |i, j   |i, j , if x i =0; |i, j   -|i, j , if x i =1; |i, j   -|i, j , if x i =1;

Example  1,1 |1, 1  +  1,2 |1, 2  +  2,1 |2, 1  +  3,1 |3,1  010 x1x1 x2x2 x3x3 Query  1,1 |1, 1  +  1,2 |1, 2  -  2,1 |2, 1  +  3,1 |3,1 

Quantum query model  Fixed starting state.  U 0, U 1, …, U T – independent of x 1, …, x N.  Q – queries.  Measuring final state gives the result. U0U0 QQ U1U1 UTUT …

Known exact algorithms

Deutsch’s problem  Determine x 1  x 2, with query access to x i.  [Cleve et al., 1998]: 1 quantum query, always the correct answer. 01 x1x1 x2x2

Dutsch-Jozsa  Distinguish whether: x 1 = x 2 =... = x N or x 1 = x 2 =... = x N or x i =0 (x i =1) for exactly ½ of i  {1, 2,..., N}. x i =0 (x i =1) for exactly ½ of i  {1, 2,..., N}.  Deterministic: N/2+1 queries.  Quantum: 1 query. x1x1 x2x2 xNxN x3x

Grover's search  Is there i:x i =1?  Promise: there is 0 or 1 i: x i =1.  Classically: N queries.  Quantum, exact: O(  N) queries. x1x1 x2x2 xNxN x3x

Exact algorithms for total functions?

Deutsch’s problem  Determine x 1  x 2, with query access to x i.  [Cleve et al., 1998]: 1 quantum query, always the correct answer. 01 x1x1 x2x2 x 1  x 2 ...  x N can be computed with N/2 queries

Montanaro et al.,  EXACT 2 4 (x 1, x 2, x 3, x 4 )=1 if there are exactly 2 i:x i =1.  Classical: 4 queries.  Quantum: 2 queries, exact. Is there a total function f(x 1,..., x N ) for which Q E (f) < D(f)/2? quantum exact deterministic

Our results

Superlinear separation  Theorem There is f(x 1,..., x N ) such that D(f)=N; D(f)=N; Q E (f)=O(N ). Q E (f)=O(N ). What should f be?

Polynomial degree lower bound  deg(f) – degree of f(x 1,..., x N ) as a multilinear polynomial.  [Nisan, Szegedy, 92, Beals et al., 98]

Basis function D(f)=3, deg(f)=2

Iterated NE 1x11x1 2x22x2 3x33x3 NE 4x44x4 5x55x5 6x66x6 7x77x7 8x88x8 9x99x9 d levels  D(f)=3 d, deg(f)=2 d

Our result  Theorem For d levels, Q E (f)=O( d ). 1x11x1 2x22x2 3x33x3 NE 4x44x4 5x55x5 6x66x6 7x77x7 8x88x8 9x99x9

Step 1  Algorithm for NE(x 1, x 2, x 3 ).  Starting state:  Result:

Step 2  p-algorithm: |  start   |  start  if f=0; |  start   |  start  if f=0; |  start   p|  start  + |  with |  |  start , if f=1. |  start   p|  start  + |  with |  |  start , if f=1. p=0  exact quantum algorithm

Step 3  p-algorithm: |  start   |  start  if f=0; |  start   |  start  if f=0; |  start   p|  start  + |  with |  |  start , if f=1. |  start   p|  start  + |  with |  |  start , if f=1.  NE(x 1, x 2, x 3 ) – 2 queries, p = -7/9 f p-algo, k queries f NE f f p’-algo, 2k queries

Step 3: result 1x11x1 2x22x2 3x33x3 NE 4x44x4 5x55x5 6x66x6 7x77x7 8x88x8 9x99x9  d levels, 3 d variables;  p-algorithm with 2 d queries. Bad p!

Step 4  Amplification f p-algo, k queries 2k queries, smaller p f Form of amplitude amplification [Brassard et al., 2000]

Final algorithm 1 level, 3 variables, 2 queries Iterate 2 levels, 9 variables, 4 queries Iterate 3 levels, 27 variables, 8 queries Amplify 3 levels, 27 variables, 16 queries...

Final result  2 11 queries for each 8 levels.  N=3 8 variables, 2 11 queries.  N=3 8k variables, 2 11k queries. Q E (f)=N

Other exact quantum algorithms

EXACT  Determine whether x i =1 for exactly k of N variables.  Montanaro et al., 2011: Algorithm: 2 out of 4, 2 queries; Algorithm: 2 out of 4, 2 queries; Computer optimization: 3 out of 6, 3 queries; Computer optimization: 3 out of 6, 3 queries; Conjecture: N/2 out of N, N/2 queries. Conjecture: N/2 out of N, N/2 queries x1x1 x2x2 xNxN x3x3

A, Iraids, Smotrovs  Exact algorithms for determining: if x i =1 for exactly N/2 i, N/2 queries; if x i =1 for exactly N/2 i, N/2 queries; if x i =1 for exactly k i, max(k, N-k) queries; if x i =1 for exactly k i, max(k, N-k) queries;  Provably optimal.  Natural computational problems;  Simple algorithms.

Algorithm: summary 1 query...

Threshold functions  Is it true that x i =1 for  k of N variables?  Exact algorithm, max(k, N-k+1) queries.  Easiest: k=N/2, N/2+1 queries.  Hardest: k=0 or k=N, N queries x1x1 x2x2 xNxN x3x3

Summary  A function that requires N queries classically, O(N ) queries for exact quantum algorithms.  First separation by more than a factor of 2.  Several other exact quantum algorithms. Advantages for exact quantum algorithms are more common that I thought

Open problems 1. d-level NE function (with 3 d variables): O( d ) query exact algorithm; O( d ) query exact algorithm; Lower bound:  ( d ). Lower bound:  ( d ). 2. Other iterated functions? 3. Other symmetric functions? 4. More exact algorithms?

Open problems 5. Lower bound methods for exact quantum algorithms? Currently known:  Bounded-error quantum lower bounds;  Q E (f)  deg(f)/2; For NE d, both of them fail.

More information  A. Ambainis. Superlinear advantage for exact quantum algorithms, arxiv:  A. Ambainis, J. Iraids, J. Smotrovs. rxiv:  A. Ambainis, J. Iraids, J. Smotrovs. Exact quantum query complexity of EXACT and THRESHOLD, arxiv: