IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 9, NO. 6, NOVEMBER/DECEMBER 2003 1495

Proposal for a Simple Quantum-Error-CorrectionTest Gate in Linear Optics

T. C. Ralph

Invited Paper

Abstract—We describe a linear quantum optical circuit capableof demonstrating a simple quantum-error-correction code in afour-photon experiment.

Index Terms—Quantum-error correction, quantum informa-tion, quantum optics.

I. INTRODUCTION

WHILE high hopes are held for the eventual demonstrationof large-scale quantum processing, present experimental

attention remains fixed on few quantum bit (qubit) demonstra-tions. Implementation of few qubit algorithms in the competingplatforms provides valuable insight into the important physicsand technical issues of different architectures.

A relatively new contender for scalable quantum computationis the scheme due to Knillet al.[1] based on single photon, dualrail qubits; linear optical networks; and photon resolving mea-surement and feedforward. Simple gates based on this schemehave been demonstrated [2] and more general gates are planned[3]–[5]. It is thus timely to consider what small-scale circuitsmight be possible with the currently available technology.

A key enabling quantum circuit is error correction [6], [7].Even medium-scale quantum processing is expected to be im-possible without error correction. Here we describe a simple ex-ample of error correction which would suit demonstration with alinear optical circuit. In its simplest form, the experiment wouldrequire only three coincident photons.

II. CIRCUIT

The quantum circuit we wish to consider is a simplification ofthe standard bit flip correcting code [8] and is shown in Fig. 1.The qubit is encoded with the following relationship betweenlogical and physical qubits

(1)

Manuscript received March 14, 2003. This work was supported by the Aus-tralian Research Council and ARDA.

The author is with the Centre for Quantum Computer Technology, Depart-ment of Physics, University of Queensland, Brisbane QLD 4072, Australia(e-mail: ralph@physics.uq.edu.au).

Digital Object Identifier 10.1109/JSTQE.2003.820943

We suppose that one of the physical qubits, say the second, suf-fers decoherence which produces random bit-flips. As a resultof this decoherence, an arbitrary initial qubit

(2)

evolves into the mixed state

(3)

where is the probability that the bit flip occurred. The aim ofthe circuit which follows is to return the qubit to its original log-ical value. To achieve this requires an ancilla photon preparedin the physical zero state. We can write the combined state ofthe qubits and ancilla after decoherence as

(4)

with the subscript “a” labeling the ancilla state. A controllednot (CNOT) gate is applied with the ancilla as target and the firstqubit as control. This transforms the state to

(5)

A secondCNOT is then applied with the ancilla as target but nowthe second qubit acts as control. The state becomes

(6)

Finally, we detect the ancilla state. If we find the ancilla in thezero state then the logical qubit is projected onto its originalstate and no correction is neccessary. On the other hand, if theancilla is found in the one state then the projected state is

(7)

1077-260X/03$17.00 © 2003 IEEE

1496 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 9, NO. 6, NOVEMBER/DECEMBER 2003

Fig. 1. Schematic of simple error-corection code.

Fig. 2. Schematic of experimental proposal. PBS: HV polarizing beamsplitter;PBS : VH polarizing beamsplitter;�BS: beamsplitter of reflectivity�; BRC:birefringent crystal;HW�: half-wave plate with rotation angle�.

We know an error has occurred which can be corrected by flip-ping the value of the second physical qubit and thus returningthe logical qubit to its initial value.

The circuit can also be understood in the language of stabi-lizer codes [9]. The code space is the1 eigenstates of ,while the error space is the1 eigenstates of . Here, is thePauli sigma z operator and indicates the tensor product of asigma z measurement on the first qubit with a sigma z measure-ment on the second qubit. The twoCNOTs achieve precisely thismeasurement with the ancilla equals zero result indicating the

1 eigenstate, and hence, the system being in the code space,while the ancilla equals one result indicates the1 eigenstateand hence, the system is in the error space and needs to be cor-rected.

The usefulness of this circuit is obviously limited by the veryspecific nature of the errors corrected, i.e., only bit flips onone of the physical qubits. Nevertheless, it exhibits the samebasic structure as more versatile codes [8] while limiting thecomplexity of the required circuit. In the next section, we willdiscuss an optical implementation of this code which appearstractable to current experimentation.

III. T HE OPTICAL IMPLEMENTATION

We consider single photon qubits with the polarization degreeof freedom determining their logical values. We define the phys-ical qubit value “zero” as being a single horizontally polarizedphoton and a physical qubit value of “one” as being

a single vertically polarized photon . Knill et al. [1]showed that nondeterministicCNOT gates could be constructedfrom linear optics with success rates of 1 in 16, using two addi-tional ancilla photons. ACNOTwould be required to produce theencoded logical qubits. Thus, an implementation of our error-correction circuit would require threeCNOTs and hence, requirethe simultaneous production of nine single-photon states. Thesuccess rate would be about 1 in 4000. Although not beyond therealm of medium term possibility, such an experiment is cur-rently not feasible. However, in the following we will discussan in principle demonstration utilizing the coincidence basiswith much lower technical requirements. The proposed setupis shown schematically in Fig. 2. The encoded state is produceddirectly using type one downconversion through a pair ofcrystals with orthogonally oriented optical axes [10]. Pairs ofphotons originating from one crystal will be horizontally po-larized whilst those originating from the other crystal will bevertically polarized. By changing the polarization of the pumpbeam, the proportion of horizontal to vertical pairs can be con-tinuously varied. In the far field, for sufficiently thin crystals,spatial information on the origin of the pairs is erased and theresulting entangled output state is approximately

(8)

where is the vacuum state, 1 and 2, label the two beams,and , and is the orientation of the

RALPH: QUANTUM-ERROR-CORRECTION TEST GATE 1497

pump beam polarization away from vertical. Coincidence de-tection will pick out only the doubly occupied parts of the state,so the effective input state is

(9)

which is logically identical to (2).Controlled decoherence can be introduced onto the second

qubit by passing it through a birefringent crystal oriented at 45degrees to horizontal [11]. The effect of the crystal is to pullapart in time the two polarization modes in the diagonal/antidi-agonal basis. When the time difference becomes an appreciablefraction of the coherence length decoherence occurs. In the hor-izontal/vertical basis the resut is random bit flips and the stateproduced can be written in the form of (3). The degree of deco-herence is a function of the crystal length.

To detect the errors we must introduce an additional ancillaqubit for readout. This can be supplied by a second downcon-verter producing just horizontal pairs. ACNOTgate which worksin the coincidence basis and does not require additional ancillamodes can be implemented using just linear optics [5], [12]. Theschematic of Fig. 2 shows the optical network needed to imple-ment the twoCNOTs in the quantum circuit (Fig. 1). The polar-ization qubits are decomposed into separate spatial modes usingpolarizing beamsplitters then the modes from different qubitsare mixed on beamsplitters. Many optical paths are possiblethrough the network but only a few result in photonic qubits atall three outputs, as determined by coincident detection of pho-tons. In such cases, quantum interference due to indistinguisha-bility of photons ensures the required transformations are im-plemented. The success rate is 1 in 81.

Correction of the decohered qubit could be implemented, ifrequired by the result of the ancilla detection, using a fast Pockelcell [13].

For most runs of the experiment, insufficient photons will bedetected and we have a null result. However, on those occasionswhen a photon is detected at the ancilla trigger and at the ancillaoutput,andphotons are detected at both qubit outputs, then to anexcellent approximation the quantum circuit of Fig. 1 will havebeen implemented. For such event, we would expect the logicalvalue of the qubits should ideally be the same as that prepared,in spite of the presence of the decohering element.

Four photon coincidences of a few per minute have beenachieved with downconversion [14]. This will be further re-duced by a factor of 1 in 81 for this proposal. On the other hand,working in the coincidence basis means, at least in principle,that this source efficiency and the efficiency of the detectorsdoes not effect the fidelity of the accepted events.

IV. CONCLUSION

We have described a simple error-correction code, and pro-posed an in-principle test of its operation using current four-photon technology. Although both the code and its implementa-tion are major simplifications over what would be required in ascalable architecture, many of the basic principles are common.We thus suggest that pursuit of experiments like the one pro-posed here will reveal much about the important physical andtechnical issues to be faced for truly scalable architectures.

ACKNOWLEDGMENT

We thank C. Ahn and G. Milburn for motivating discussions.

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T. C. Ralph graduate from Macquarie University, Sydney, Australia, and re-ceived the Ph.D. from the Australian National University, Canberra ACT, Aus-tralia.

He currently holds a Queen Elizabeth II Fellowship at the University ofQueensland, Queensland, Australia, and is a Research Manager in the Centrefor Quantum Computer Technology.