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
0 1 0 ... 0
x1 x2 x3
xN




Is there i:xi=1?
Classically, N queries required.
Quantum: O(N) queries [Grover, 96].
Quantum, exact: N queries.
Model
Query model
f(x1, ..., xN), xi{0,1}.
 xi given by a black box:
 Function
i
Complexity = number of queries
xi
Queries in the quantum world
states: |1,1, |1, 2, …, |N, M.
 Query:
 Basis


|i, j  |i, j, if xi=0;
|i, j  -|i, j, if xi=1;
Example
x1 x2 x3
0 1 0
1,1|1, 1+1,2|1, 2+2,1|2, 1+3,1|3,1
Query
1,1|1, 1+1,2|1, 2- 2,1|2, 1+3,1|3,1
Quantum query model
U0
 Fixed
Q
U1
…
Q
UT
starting state.
 U0, U1, …, UT – independent of x1, …, xN.
 Q – queries.
 Measuring final state gives the result.
Known exact algorithms
Deutsch’s problem
0 1
x1 x2
 Determine x1x2, with query access to xi.
 [Cleve et al., 1998]: 1 quantum query,
always the correct answer.
Dutsch-Jozsa
0 1 0 ... 0
x1 x2 x3
xN
 Distinguish


whether:
x1 = x2 = ... = xN or
xi=0 (xi=1) for exactly ½ of i{1, 2, ..., N}.
 Deterministic:
N/2+1 queries.
 Quantum: 1 query.
Grover's search
0 1 0 ... 0
x1 x2 x3
xN




Is there i:xi=1?
Promise: there is 0 or 1 i: xi=1.
Classically: N queries.
Quantum, exact: O(N) queries.
Exact algorithms for total
functions?
Deutsch’s problem
0 1
x1 x2
 Determine x1x2, with query access to xi.
 [Cleve et al., 1998]: 1 quantum query,
always the correct answer.
x1x2...xN can be computed
with N/2 queries
Montanaro et al., 2011.
 EXACT24(x1, x2, x3, x4)=1
if there are
exactly 2 i:xi=1.
 Classical: 4 queries.
 Quantum: 2 queries, exact.
Is there a total function f(x1, ..., xN) for
which QE(f) < D(f)/2?
quantum exact
deterministic
Our results
Superlinear separation
 Theorem


There is f(x1, ..., xN) such that
D(f)=N;
QE(f)=O(N0.86...).
What should f be?
Polynomial degree lower bound
– degree of f(x1, ..., xN) as a
multilinear polynomial.
 [Nisan, Szegedy, 92, Beals et al., 98]
 deg(f)
Basis function
D(f)=3, deg(f)=2
Iterated NE
NE
NE
NE
x1
x2
x3
x4
x5
NE
x6
x7
x8
x9
d levels  D(f)=3d, deg(f)=2d
Our result
NE
NE
NE
x1
x2
 Theorem
x3
x4
x5
NE
x6
x7
x8
x9
For d levels, QE(f)=O(2.593...d).
Step 1
Algorithm for NE(x1, x2, x3).
Starting state:
Result:
Step 2
 p-algorithm:


|start  |start if f=0;
|start  p|start + | with ||start, if f=1.
p=0  exact quantum algorithm
Step 3
 p-algorithm:


|start  |start if f=0;
|start  p|start + | with ||start, if f=1.
 NE(x1, x2, x3) – 2 queries, p = -7/9
NE
f
p-algo, k queries
f
f
f
p’-algo, 2k queries
Step 3: result
NE
NE
NE
x1
x2
x3
x4
x5
NE
x6
 d levels, 3d variables;
Badd p!
 p-algorithm with 2 queries.
x7
x8
x9
Step 4
Amplification
f
p-algo, k queries
f
2k queries, smaller p
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
 211
queries for each 8 levels.
 N=38 variables, 211 queries.
 N=38k variables, 211k queries.
QE(f)=N0.86...
Other exact quantum
algorithms
EXACT
0 1 0 ... 0
x1 x2 x3
xN
 Determine
whether xi=1 for exactly k of N
variables.
 Montanaro et al., 2011:



Algorithm: 2 out of 4, 2 queries;
Computer optimization: 3 out of 6, 3 queries;
Conjecture: N/2 out of N, N/2 queries.
A, Iraids, Smotrovs
 Exact


algorithms for determining:
if xi=1 for exactly N/2 i, N/2 queries;
if xi=1 for exactly k i, max(k, N-k) queries;
 Provably
optimal.
 Natural computational problems;
 Simple algorithms.
Algorithm: summary
1 query
1 query
...
...
Threshold functions
0 1 0 ... 0
x1 x2 x3
xN
it true that xi=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.
 Is
Summary
 A function
that requires N queries
classically, O(N0.86...) 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
d-level NE function (with 3d variables):
1.


2.
3.
4.
O(2.593...d) query exact algorithm;
Lower bound: (2.11...d).
Other iterated functions?
Other symmetric functions?
More exact algorithms?
Open problems
Lower bound methods for exact quantum
algorithms?
Currently known:
5.
Bounded-error quantum lower bounds;

QE(f)  deg(f)/2;
For NEd, both of them fail.

More information
 A. Ambainis.
Superlinear advantage for
exact quantum algorithms,
arxiv:1211.0721.
 A. Ambainis, J. Iraids, J. Smotrovs. Exact
quantum query complexity of EXACT and
THRESHOLD, arxiv:1302.1235.