Quantum algorithms in the presence of decoherence:

optical experiments

Masoud Mohseni, Jeff Lundeen, Kevin Resch and Aephraim Steinberg

Department of Physics, University of Toronto

Friendly neighborhood theorists: Daniel Lidar, Sara Schneider,...Helpful summer student: Guillaume Foucaud

MotivationPhotons are an ideal system for carrying quantum info.

(Nonscalable) linear-optics quantum computation may prove essential as part of quantum communications links.

Efficient (scalable) linear-optical quantum computation isa very promising avenue of research, relying on the same toolbox (and more).

In any quantum computation scheme, the smoky dragonis decoherence and errors.

Without error correction, quantum computationwould be nothing but a pipe dream.

We demonstrate how decoherence-free subspaces (DFSs) maybe incorporated into a prototype optical quantum algorithm.

Prototype algorithm: Deutsch's Problem (2-qbit version)

An oracle takes as input a bit x, and calculates an unknownone-bit function f(x).

[quantum version: inputs x&y; outputs x & y f(x)]

Our mission, should we decide to accept it:Determine, with as few queries as possible, whether or not f(0) = f(1).

Classically: must measure both f(0) and f(1).[For n-bit extension, need at least 2n-1+1 queries]

Quantum mechanically: a single query suffices.[Even for n-bit problem, since only yes/no outcome desired.]

Standard Deutsch-Jozsa Algorithm

H

H

x x H0

1 y y⊕f(x)

)1(f)0(f ⊕

2

10 −

Physical realization of qubits

We use a four-rail representation of our two physical qubits and encode the logical states 00, 01, 10 and 11 by a photon traveling down one of the four optical rails numbered 1, 2, 3 and 4, respectively.

1234 0001

0010

0100

1000

Photon number basis

1st qubit 2nd qubit

11

10

01

00

Computational basis

Bob (oracle)Alice Alice

DJ algorithm and 4-rail qubits

[Cf. Cerf, Adami, & Kwiat, PRA 57, R1477 (1998)]

Quantum gate Four rails implementation

50/50 beam splitters swap between two rails

NOT-2

X

00

11

10

01NOT-1 00

11

10

01X

CNOT-1 00

11

10

01CNOT-2 00

11

10

01

Hadamard-200+01

10-11

10+11

00-01

H

Hadamard-100+10

01-11

00-10

01+11H

00

11

10

01

00

11

10

01

00

11

10

01

00

11

10

01

00

11

10

01

00

11

10

01

Quantum gate Four rails implementation

It is easy to implement a universal set up of one and two qubit operations in such a representation

Implementation of simple gates

Balanced oracle-01f(0)=0,f(1)=1 00

11

10

01

00

11

10

01

00

11

10

01

Constant oracle-00 f(0)=f(1)=0 00

11

10

01

00

11

10

01

X

Constant oracle-11 f(0)=f(1)=1 00

11

10

01

00

11

10

01

XX

Balanced oracle-10f(0)=1,f(1)=0 00

11

10

01

The transformations introduced by the 4 possible functions or “oracles” can also be implemented in this representation.

Implementation of the oracle

2/)t()(i 2z2

z1e ϕσ−σ

00 0001 eiϕ 01

11 1110 eiϕ 10

But after oracle, only qubit 1 is needed for calculation.

Encode this logical qubit in either DFS: (00,11) or (01,10).

Error model and decoherence-free subspaces

Consider a source of dephasing which acts symmetricallyon states 01 and 10 (rails 2 and 3)…

Modified Deutsch-Jozsa Quantum Circuit

1 y⊕f(x)

H x x H0

yH

DFSs: see Lidar, Chuang, Whaley, PRL 81, 2594 (1998) et cetera.Implementations: see Kwiat et al., Science 290,498 (2000)

and Kielpinski et al., Science 291, 1013 (2001).

Schematic diagram of D-J interferometer

1 2 3 4

1

2

3

4

1 2 3 4

Oracle00

01

10

11

“Click” at either det. 1 or det. 2 (i.e., qubit 1 low)indicates a constant function; each looks at an interferometer comparing the two halves of the oracle.

Interfering 1 with 4 and 2 with 3 is as effective as interfering

1 with 3 and 2 with 4 -- but insensitive to this decoherence model.

3

4

1

2

1

2

4

23

Experimental Setup

Oracle

Swap

Preparation

Random Noise

Mirror

Waveplate

Phase Shifter

PBS

Detector

2/λ

A

B

C

D

34

34

DJ experimental setup

C B CC CBB B

DFS Encoding Original encoding

Constant function

Balanced function

C

B

DJ without noise -- raw data

C B CC CBB B

DFS Encoding Original encoding

C

B

Constant function

Balanced function

DJ without noise -- results

C B CC CBB B

DFS Encoding Original Encoding

C

B

Constant function

Balanced function

DJ with noise-- results

Coming Attractions:Non-orthogonal State Discrimination

• Non-orthogonal quantum states cannot be distinguished with certainty.

• This is one of the central features of quantum information which leads to secure (eavesdrop-proof) communications.

• Crucial element: we must learn how to distinguish quantum states as well as possible -- and we must know how well a potential eavesdropper could do.

H-polarized photon 45o-polarized photon

(work with J. Bergou et al.)

Theory: how to distinguish non-orthogonal states optimally

Step 1:Repeat the letters "POVM" over and over.

The view from the laboratory:A measurement of a two-state system can onlyyield two possible results.

If the measurement isn't guaranteed to succeed, thereare three possible results: (1), (2), and ("I don't know").

Therefore, to discriminate between two non-orth.states, we need to use an expanded (3D or more)system. To distinguish 3 states, we need 4D or more.

Step 2:Ask Janos, Mark, and Yuqing for help.

Experimental layout

(ancilla)

Success!

The correct state was identified 55% of the time--Much better than the 33% maximum for standard measurements.

"I don't know"

"Definitely 3"

"Definitely 2"

"Definitely 1"

Summary• We have demonstrated the utility of decoherence-free subspaces in a prototype linear-optical quantum algorithm.

• The introduction of localized turbulent airflow produced a type of “collective” optical dephasing, leading to large error rates.

• With the DFS encoding, the error rate in the presence of noise was reduced to 7%, essentially its pre-noise value.

• We note that the choice of a DFS may be easier to motivate via consideration of the physical system than from purely theoretical (quantum circuit) considerations!

• More recent results: successfully distinguish among 3 non-orthogonal states 55% of the time, where standard quantum measurements are limited to 33%. Also: "state filtering" or discrimination of mixed states.