Efficient Discrete-Time Simulations of Continuous-Time Quantum Query Algorithms

QIP 2009January 14, 2009

Santa Fe, NM

Rolando D. SommaJoint work with R. Cleve, D. Gottesman, M. Mosca, D. Yonge-Mallo

Query or Oracle Model of Computation

Given a black-box (BB)

€

x j ∈ {0,1} BB

€

j →

€

0 ≤ j ≤ N −1

For quantum algorithms, we consider a reversible version of BB:

€

j →

€

→ (−1)x j j

€

QX j = (−1)x j j

€

QX

Want to learn a property of the N-tuple

€

x0, x1,..., xN−1( )

Query or Oracle Model of Computation

Oracle models are useful to obtain bounds in complexity and to make a fair comparisson between quantum and classical complexities

Quantum algorithms in the oracle model

U1 U2 U3 …

€

0

€

0

€

M

Known unitaries

Output gives some property of

€

x0,x1,K ,xN−1[ ]MMMMM

M

€

QX

€

QX

€

QX

Examples* Shor’s factorization algorithm: Period-finding

* Grover’s algorithm: find a marked element

* Element Distinctness (Ambainis): finding two equal items

€

f ( j + r) = f ( j) = x j ; T ≈ log3(N)

€

j / x j =1 ; T ≈ N

€

i, j / x i = x j ; T ≈ N 2 / 3

Continuous-Time Quantum Query Model of Computation

2- Time-dependent Driving Hamiltonian (known)

3- Evolution time (or total query cost) T>0

€

HD (t)

€

0

€

0

€

MOutput gives

some property of

€

x0, x1,K , xN−1[ ]

€

U = exp −i (HX + HD (t))dt0

T

∫ ⎡

⎣ ⎢

⎤

⎦ ⎥

MMMMM

M

€

j →

€

→ e-iH Xθ j

1- Query Hamiltonian

€

HX j = x j j

€

e−iH X π j = QX j = (−1)x j j

QXθ j = e−iHXθ j

€

QXθ

Query cost

fractional query

* E. Farhi and S. Gutmann, Phys. Rev. A 57, 2043 (1998)

Motivations:

•Some quantum algorithms have been discovered in the continuous time query model

“Exponential algorithmic speed up by quantum walk”, Childs et. al.[Proc. 35th ACM Symp. On Th. Comp. (2003)]

Given: an oracle for the graph, and the name of the Entrance.Find the name of the Exit.

Motivations:

•Some quantum algorithms have been discovered in the continuous time query model

“A Quantum Algorithm for Hamiltonian NAND tree”, Farhi, Goldstone, Gutmann quant-ph/0702144

The query Hamiltonian is built from the adjacency matrix of a graph determined by the tree and the input state. It outputs the (binary) NAND in time

€

N

N

Motivations:

Is it possible to convert a quantum algorithm in the CT setting to a quantum algorithm in the more conventional query model?

We present a method to do it at a cost

€

T /ε( ) log(T /ε)

Yes:

It has been known(2) that this can be done with cost

€

eη HD T( )1+1/η

(2) D. Berry, G Ahokas, R. Cleve, and B.C. Sanders, Commun. Math. Phys. 270, 359 (2007)

Q(1): Is the CT query model more powerful than the conventional query model?

The actual implementation of a quantum algorithm in the CT setting may require knowledge on the query Hamiltonian which my not be an available resource.

(1) C. Mochon, Hamiltonian Oracles, quant-ph/0602032

MAIN RESULTS:

Theorem: Any continuous-time T-query algorithm can be

simulated by a discrete-time O(T log T )-query algorithm

Corollary: Any lower bounds on discrete query complexity carry over to continuous query complexity within a log factor

Quantum Algorithm: Overview

• Step 1: Discretization using a (first order) Suzuki-Trotter approximation

• Step 2: Probabilistic simulation of fractional queries using (low-amplitude) controlled discrete queries 1 and 2 yield simulations of cost O(T2)

•Step 3: Reduction on the amount of discrete queries by disregarding high- Hamming weight control-qubit states

• Step 4: Correction of errors due to step 2

The construction has many steps…

Step 1: Trotter-Suzuki Approximation

€

0

€

0

€

MOutput gives

some property of

€

x0, x1,K , xN−1[ ]

€

U = exp −i (HX + HD (t))dt0

T

∫ ⎡

⎣ ⎢

⎤

⎦ ⎥

Algorithm in the CT setting

MMMMM

U1 U2 U3…

€

0

€

0

€

M

€

QXθ

€

QXθ

€

QXθ = e−iH Xθ

U j = e−iH D ( t j )θ

€

θ =T

p=

ε1

2Tr;

p =2T 2r

ε1

€

F ≥1−ε1

Step 1: Fidelity

€

r =1

THD (t) dt

0

T

∫

Still p>>T fractional queries

MMMMM

Step 1: Trotter-Suzuki Approximation

U1 U2 U3…

€

0

€

0

€

M

€

QXθ

€

QXθ

€

QXθ = e−iH Xθ

U j = e−iH D ( t j )θ

€

θ =T

p=

ε1

2Tr;

p =2T 2r

ε1

€

F ≥1−ε1

Step 1: Fidelity

€

r =1

THD (t) dt

0

T

∫

Still p>>T fractional queries

MMMMM

€

QXθ

€

QX

€

QX€

+

€

e−iθ − −

It doesn’t work in general…

Step 2: Probabilistic Simulation of Fractional Queries

€

QXθ = e−iH Xθ = e−iθ / 2 cos(θ /2)1+ isin(θ /2)QX[ ]

€

QX = e−iH X π

€

0 → QXθ Pr. success : ps ≥1−θ

1 → QX−π / 2 Pr. failure : p f ≤ θ

€

QXθ

€

QX

R1 R2

€

0 M

€

cos(θ /2) 0 + i sin(θ /2)

€

R2 :0 a cos(θ /2) 0 + sin(θ /2) 1

1 a sin(θ /2) 0 − cos(θ /2) 1

⎧ ⎨ ⎪

⎩ ⎪

Why do we want this conversion?

€

p =2T 2r

ε1

>> T;

θ =T

p=

ε1

2Tr<<1

The actual query cost is much lower than p.In step 3, we take advantage of this situation.

Step 2: Probabilistic Simulation of Fractional Queries

U1 U2 U3…

€

0

€

0

€

M

€

QXθ

€

QXθ

MMMMM

€

0

U1 U2 U3…

€

0

€

0

€

MMMMMM

R1

R1

R1

M

M

M

€

QX

€

QX

R2

R2

R2

€

We succeed if

00...0

€

PS = ps( )p

<<1

€

M

€

QX

€

0

€

0

€

p =2T 2r

ε1Up

€

QXθ

Step 3: Reducing the amount of queries

m queries

€

P'S = ps( )m

≥ (1−θ)m = 3/4 For a segment of size m,it is likely to succeed

€

m =1

4θ=

Tr

2 ε1

<< p =2T 2r

ε1

There are 4T segments of that sizein the total circuit

We break the circuit in segments of size m :

€

0

U1 U2 U3…

€

0

€

0

€

MMMMMM

R1

R1

R1

M

M

M

€

QX

€

QX

R2

€

We succeed if

00...0

€

PS = ps( )p

<<1

€

M

€

QX

€

0

€

0 R2

R2€

0 R1 MR2…

€

M

€

M

Step 3: Reducing the amount of queries

U1 U2 U3…

€

0

€

0

€

M

R1

R1

R1

M

M

M

€

QX

€

QX

R2

R2

R2€

Success if 0⊗m

€

P'S ≥ 3/4

m queries

m

€

∝ cos(θ /2) 0 + i sin(θ /2) 1( )⊗m

Density of states

€

M

Hamming weight

Poisson distribution: Exponential decay

€

Expected value of HW : A ≈ mθ ≈1/4

Um

€

QX

€

m =1

4θ

Step 3: Reducing the amount of queries

U1 U2 U3…

€

0

€

0

€

M

R1

R1

R1

€

QX

€

QX

€

M

m queries

m

€

QX

Density of states

Average: A<1/2

Hamming weight cutoff

€

k ∈ O log(T /ε2)[ ]

€

Fidelity :

F ≥1−ε2

At most k<<m full queries are needed !

Step 3: Reducing the amount of queries

U1 U2 U3 …

€

0

€

0

€

M

R1

R1

R1

€

QX

€

QX

€

Mm

€

QXm full queries

V2 V3 …

€

0

€

0

€

M

R1

R1

R1

€

QX

€

QX

€

Mm

€

QX Vk

full queries

U1

€

k ∈ O(log(T /ε2)) << m

Step 3: Reducing the amount of queries

Vj

m

Asks the value of the Hamming weight

Implements the desired sequence of U’s

V2

m

Example:

€

If in 00...0

€

Implement U2 , U3 , ..., Um

U2 U3 Um

€

M

€

M

€

M

…

Step 3: Reducing the amount of queries

€

So far, if we replace the circuit as explained in step 3,

the total amount of full queries is ( p /m)log(T /ε2)∝ T /ε1( )log(T /ε2)

€

However, the probability of success is still

PS = pS( )p

≈ (1−θ)p <<1

We build Step 4 to error correct and increase the probability of success towards 1

Step 4: Error correction

U1 U2 U3…

€

0

€

0

€

M

R1

R1

R1

€

QX

€

QX

€

M

m queries

m

€

QX€

We succeed if

0⊗m

€

P'S = ps( )m

≥1/4M

M

M

€

A failure (i.e., 1 ) is equivalent to perform the

operation QXπ / 2 instead of the desired operation QX

θ

X

€

QXθ Um ...QX

−π / 2U j ...QXθ U2QX

θ U1

1- We undo the circuit:

2- We redo it:

€

QXθ Um ...U j +1QX

−π / 2( )

+

€

QXθ Um ...QX

θ U j +1

Step 4: Error correction

1- We undo the circuit:

2- We redo it:

€

QXθ Um ...U j +1QX

π / 2( )

+

€

QXθ Um ...QX

θ U j +1

Both, the undoing and redoing parts require the simulation of fractional queries with phases ± . Therefore, to reduce the total amount of queries, each of these

operations have to be simulated probabilistically as explained in step 2.

This yields a branching process, in which we iterate the error correction procedure. In the worst case, the undoing and redoing parts succeed

(each) with probability bounded below by 3/4.

Step 4: Error correction

The probability of success increases towards 1 exponentially fast with k

Step 4: Error correction

Each of the circuits in the branching process (of size m or smaller) is simulated using the “trick” of step 3 to reduce the amount of queries

Because the size of the tree associated to the branching process is a O(1) constant, to succeed with probability (say) 1- 3 , we need to simulate O(T/ 3) circuits ONLY.

Complexity of the simulation

For fidelity 1-, our simulation requires full queries

€

O[log(T /ε)(T /ε)]

€

O[log(T /ε)T log(1/ε)]For classical input/output, the overall complexity is

€

We choose ε1 , ε2 , ε3 , which correspond to the errors from

the Trotter approximation, Hamming weight cutoff, and the prob.

of failure in the simulation of the branching process, of order ε

Step 5: Conclusions!

€

Described an efficient discrete query simulation of continuous- time query algorithms

with complexity O(T log(T /ε)log(1/ε)), for fidelity 1- ε

€

The amount of known unitaries required is of order O(rT 2 log(T /ε)),

but this is not essential (depends on behavior of driving Hamiltonian)

Improvements?