Linear Optical Quantum Computing
Anna Dardia
I. INTRODUCTION
There are a wide range of proposed systems with which quantum computing could be physically realized, including atom and ion-trap quantum computing, nuclear magnetic resonance, and quantum dots. Quantum computing with linear optics uses photons as qubits. Photons have the advantage of minimal decoherence. However, photons do not natrually interact with one another, so in order to make two-qubit gates an effective interaction needs to be induced. In this paper, we will discuss the construction of quantum gates using linear optics, how multi-qubit gates can be constructed, as well as the Knill-Laflamme-Millburn (KLM) protocol for linear optical quantum computing (LOQM). Some of the error correction methodology will be discussed, as well as sources of error arising from the physical linear optical components and resources. Finally, we briefly discuss some developing improvements on the KLM regime.
II. A BRIEF BACKGROUND ON OPTICS
The energy of a classical electromagnetic field is Z 1 X hωk H = (E2 + B2)dr = a a∗ 2 2 k k V k
∗ The electromagnetic field is quantized by turning the coefficients ak and ak into operators with commutation relations
† [a ˆi , aˆj ] = δij † † [a ˆi, aˆj] = [a ˆi , aˆj ] = 0
Letting
1 rmω 1 aˆ = √ xˆ + i √ ˆp 2 ¯h m¯hω The energy can now be expressed in the form of a harmonic oscillator
pˆ2 mω2xˆ2 Hˆ = + 2m 2 The eigenstates of Hˆ are called Fock states, labelled |ni .aˆ anda ˆ† are creation and annihilation operators acting on the Fock states as follows √ √ aˆ |ni = n |n − 1i , aˆ† |ni = n + 1 |n + 1i
B. Linear Optics
Optical components are linear if their output modes are linear combinations of their input modes.
ˆ† X † bj = Mjkaˆj k Basic linear optical components are beamsplitters, half and quarter-waveplates and phase-shifters. These components can be described mathematically by unitary operations. 2
A single-mode phase shift changes the phase of the electromagnetic field in a particular mode. It’s interaction Hamiltonian is given by ˆ † Hφ = φaˆinaˆin using the convention ¯h = 1. This Hamiltonian commutes with the number operator, and so the total photon number is a conserved quantity.
FIG. 1. Diagrams of a beamsplitter (a) and a polarizing beamsplitter (b) with incoming and outgoing modes depicted [1]
In general, the interaction Hamiltonion of a beamsplitter can be written
iφ † ˆ −iφ ˆ† HBS = θe aˆinbin + θe aˆinbin This operator also commutes with the number operator, and the number of photons is again a conserved quantity. This description also applies to polarization evolution due to the action of a waveplate. Rather than spatial modes, incoming modes have different polarization, and the angles θ and φ refer to angles of rotation. We may also construct polarizing beamsplitters, often cut to separate horizontal and vertical polarization. In this case, the transformation is as follows ˆ aˆin,H → aˆout,H , aˆin,V → bout,V
ˆ ˆ ˆ bin,H → bout,H , bin,V → aˆoutV Where the input and output modes are as depicted in Fig 1.
III. QUBITS IN LINEAR OPTICS
Th e qubits are quantum systems with SU(2) symmetry. In the case of LOQC, the typical qubit is a photon that can be in one of two modes. Qubits are encoded with dual rail logic, in this case with respect to spatial modes. We define logical zero and one:
|0iL ≡ |1ia ⊗ |0ib
|1iL ≡ |0ia ⊗ |1ib
Where a denotes one spatial mode and b denotes another. It is mathematically equivalent to use polarization instead of spatial modes, defining |0iL = |Hi , |1iL = |V i. While the two are equivalent, going forward this paper will be concerning mostly dual rail logic with spatial modes.
IV. SINGLE-QUBIT GATES
In order for a physical system to be capable of quantum computation, we require a universal set of gates [2]. We will first discuss single-qubit operations, which are generated by the Pauli operators σx, σy, σz, as any single-qubit 3 unitary can be decomposed into rotations in the Bloch sphere. As such, all single-qubit gates can be implemented using only beamsplitters and phase shifters. Rotations about the Z axis can be accomplished using a phase shifter. Consider the application of a phase shifter on the first mode of the qubit α |0iL + β |1iL:
α |0iL + β |1iL = α |01i + β |10i → α |01i + βeiφ |10i iφ/2 −iφ/2 iφ/2 = e (e α |0iL + e β |1iL) iφ/2 = e RZ (φ)(α |0iL + β |1iL)
Up to an arbitrary phase factor, a rotation about the Z axis of the Bloch sphere has been performed.
Rotations about the Y axis can be performed by a beamsplitter. A rotation of 2θ requires a beamsplitter with angle θ, and φ = 0, as the action of a beamsplitter on a photon in modes a and b is
aˆ† → (cos(θ)ˆa† + sin(θ)ˆb†) ˆb† → (−sin(θ)ˆa† + cos(θ)ˆb†)
Let us consider the action of the beamsplitter on an arbitrary qubit α |0iL + β |1iL:
α |0iL + β |1iL = α |01i + β |10i → α(cos(θ) |01i − sin(θ) |10i) + β(cos(θ) |10i + sin(θ) |01i) = cos(θ)(α |01i + β |10i) − sin(θ)(α |10i − β |01i)
iθYq = e (α |0iL + β |1iL)
= RY (−2θ)(α |0iL + β |1iL)
We can see that a rotation of −2θ about Y has been accomplished. From here we may construct any deterministic single-qubit gate.
V. THE KLM PROTOCOL
Multi-qubit gates are difficult to construct with photons, as photons do not interact with one another- actually we cannot construct deterministic two-qubit gates using linear optics. In 2001, Knill, LaFlamme, and Millburn constructed a protocol to create universal quantum computers using linear optics. This protocol addressed the difficulty of constructing multi-qubit gates by performing probabilistic operations offline, and then boosting the probability of success using teleportation [3]. The gate we will focus on is the controlled-phase sign gate (CZ).
Control Target CZ |0i |0i |00i |0i |1i |01i |1i |0i |10i |1i |1i -|11i
q1q2 This transformation can be expressed as |q1q2i → (−1) |q1q2i and can be constructed using two nonlinear-sign (NS) gates [3]. The NS gate transforms the three lowest Fock states as
α |0i + β |1i + γ |2i → α |0i + β |1i − γ |2i
Consider Fig 2 with input |φ1i ⊗ |φ2i = (α |01i + β |10i)(γ |01i + δ |10i), which is separable. The NS gates impart a phase shift π, and after the second beamsplitter the output is αγ |00i + αδ |01i + βγ |10i − βδ |11i, which is entangled. While an NS gate cannot be constructed with linear optics, the gate can be constructed probabilistically with the 4
FIG. 2. CZ gate constructed from two NS gates and two beamsplitters [1]
1 use of projective measurements [1]. The maximum probability of success for an NS gate is 4 , which means that the 1 probability of success of a CZ gate constructed from two NS gates will be at most 16 . This, of course, is insufficient for scalable LOQM, as it is very likely that the gate will fail, overwhelmingly likely as the number of probabilistic gates increase. The solution is then to remove the probabilistic gate from the circuit, prepare it as an off-line resource, and then ”teleport” the gate into the circuit. That is, the CZ gate is applied offline (probabilistically), and once we know that the gate was successful, the qubits are teleported into the circuit and the information remains uncorrupted.
FIG. 3. Teleportation using a QFT[1]
We are interested then in teleporting an arbitrary state α |0i + β |1i, as linearity ensures that if this is possible, any superposition of the arbitrary state is also possible. This is accomplished using an n+1 point discrete quantum Fourier transform (QFT)(Which can also be seen as a generalized 50:50 beamsplitter [4]). We note that the QFT erases path information of incoming modes. In the quantum channel we have the general state
n 1 X j n−j j n−j |tni = √ |1i |0i |0i |1i n + 1 j=0
We can write the QFT in matrix notation as
1 (i − 1)(j − 1 (F ) = √ exp 2πi n ij n n
The QFT is applied to the input and to the first n modes of |tni. The number of photons m in the output mode are counted, and the state is teleported to the n+m mode of the quantum channel. Measuring m photons means either the input mode did not have a photon, and the m photons came from |tni, or the input mode did have a photon, and the other photons originated from |tni. But by construction, the m mode of the remaining |tni modes must have the same number of photons as the input mode. The erasure of incoming mode path information means that these two possibilities are added. Thus, the qubit has been teleported to the n+m mode. There is still a chance of failure- if 0 5 photons are counted in the output, the state collapses to |0i, and if n+1 photons are counted, the state collapses to n 2 |1i. But this chance is far smaller than that of the probabilistic gate alone; the probability of success is then ( n+1 ) for the teleportation of two qubits. This chance of failure may be further minimized through error correction. We should note the complexity of the required auxiliary states. For example, the CZ gate requires the auxiliary state n 1 X i n−i i n−i j n−j j n−j |CZ i = (−1)(n−i)(n−j) |1i |0i × |0i |1i |1i |0i |0i |1i (1) n n + 1 i,j=0 This is rather complicated, and certainly quite resource-demanding to create. However, some error correction protocols may make this less costly.
VI. ERRORS AND ERROR CORRECTION
A. KLM: Parity Encoding
We previously discussed the case of probabilistic gate failure, occurring in the cases where 0 or n+1 photons are measured in the output, as this represents a measurement of the input. KLM introduced parity encoding in order to allow the qubit to survive gate failure. the parity encoding involves defining logical zero as a superposition of even parity states, while the logical one is defined as a superposition of odd parity states:
|0iL ≡ |HHi + |VV i
|1iL ≡ |HV i + |VHi
If an arbitrary qubit α |0iL + β |1iL is measured with the result H, the logical qubit is projected to α |Hi + β |V i. Instead of being destroyed, the encoding is just reduced from parity to polarization. If the measurement returns V, the qubit is projected to α |V i + β |Hi, which preserves the superposition but with a bit-flip. However, the result V informs us that this has occurred, and can be corrected. Let us consider the case of the teleportation of a parity qubit with t1 teleporter. If the probabilistic gate succeeds, the qubit is teleported, and corrections can be applied based on the measurement result of the value of the remaining polarization qubit. If the gate fails, the photon measurement result (0 or m+n photons) can be used to correct the remaining polarization qubit. In short, we get to try again.
B. KLM: Concatenation of the Code
Concatination is used to boost the size of the code in order to increase the probability of success. At the first level we have four-photon encode states [3] 4 |0iL = |00iL + |11iL 4 |1iL = |10iL + |01iL
We can do this with 8 photons at the second level of concatenation, and so forth. With this, we may construct teleportation circuits that are more likely to succeed.
To use these encoded qubits, we require a universal set of gates. For rotations about the X axis, we may use the operation Xθ = cos(θ/2) − isin(θ/2) by applying it to a√ polarization qubit. However, we also need to be able to perform π/2 rotations about Z (using the operation Zθ = 1/ 2(I − iZ)), which requires applying Zπ/2 on each qubit and then applying a CZ-gate between the qubits. But as we discussed, CZ-gates in LOQC are probabilistic, so we will need to use the teleportation scheme as the Zπ/2 operation will be probabilistic as well. This means there are more chances of gate failure, and the larger the optical system, the larger this error will be.
Additionally, it is not easy to perform CZ-gates with high probabilities of success just by using concatenation as our error prevention method. It was demonstrated in [5] that in order to perform a CZ gate with 95 percent probability of success, we require first order concatenation, and t3 teleporters. This requires a large number of operations, and while the KLM scheme is scalable in terms of physical resources, it is also complicated and costly. 6
C. Errors in Optical Components
1. Photon Detectors
Linear optical quantum computing schemes require photon detectors to gain information about the state of interest. There are two main types of photon detectors: detectors with single-photon resolution (number-resolving detectors), and detectors which distinguish between ’no photons’ and ’photons’ (bucket detectors). The KLM scheme assumes and relies upon number-resolving detectors. Up until this point, we have assumed the detectors are capable of perfect single-photon resolution. However, real photon detectors can sometimes fail to detect photons (photon loss), or register more counts than the number of photons (dark counts). While we cannot make statements about the ”real” number of photons in the input state, as prior to measurement this is not a well-defined quantity, we can consider the efficiency of the detector itself. The detector efficiency is the probability that a single photon input will be registered as a count by the detector. Dark counts can be thought of as the probability that a vaccuum state will register as a detector count. In current use in LOQM are Avalanche Photo-Diodes (APDs), which have an efficiency around 85 percent for wavelengths of 660nm [1]. Dark counts can be minimized by operating the detectors at low temperatures. Currently the development of number-resolved detectors is underway. One technique is using a superconducting transition-edge sensor, which measures the rise in temperature of absorbers due to the incoming photons. This requires low temperatures, but can reach efficiencies of 88 percent with very few dark counts.
Correcting for photon loss is not easy. Photon loss, or the failure to measure a photon, can also be thought of as a single qubit failure, as in LOQC single-qubit measurements amount to the detection of photons. This means that every logical qubit needs to consist of multiple photons if we want to recover from photon loss. One proposed error correction method involves redundant encoding such that logical qubits are encoded in GHZ states of parity encoded qubits, briefly discussed in [1]. In any photon detector however, we note that beams which are not perfectly collimated will result in error. This is due to the fact that in the Coulomb gauge, polarization and direction of propogation are orthogonal. So the plane of detection must be perpindicular to the Poynting vector k. Realistic beams have small angular spreads, and as a result some modes of the beam will encounter the detector at an angle. This is a fundamental detection error which may be minimized but not eliminated.
2. Photon Sources
We also thus far have assumed perfect single-photon sources. However, single photons cannot be localized as single massive particles can be. Let us consider a 1-D cavity of length L. The possible wave vectors kn are given by kn = nπ/L, with frequencies ωn. Let us measure time in units of Lπ/c, in order to denote ωn = n. The positive-frequency component of the field is
∞ X −int aˆ(t) = aˆne n=1 Where the creation and annihilation operators are those discussed in Sec I. Placing the detector at the origin, the probability of detection per unit time is
p(t) = µn(t)
Where n(t) is the number operator, defined by n(t) =
∞ X † |1; fi = fmaˆm |0i m=1
P∞ 2 Where |0i is the global vacuum state and m=0 |fm| = 1. We then arrive at our counting probability[1] ∞ 2 X −ikt n(t) = fke
m=0 7
Which is a periodic function of time with period 2π. The function will look like a somewhat narrow peak (the width depending on parameters such as the cutoff frequency, etc.) We cannot obtain perfect resolution of single photons. We must also be careful to select sources which have low probabilities of emitting multiple photons at once.
3. Circuit Components
The physical design of the optical components which compose the quantum gates may also be sources of error. For example, beamsplitters and waveplates are usually made of media which have a nonzero absorption probability. This can cause photon loss, which will add up in the case of large circuits. Additionally, birefringence in the component media can cause photoemission and squeezing.
Because probabilistic gates often require multiple attempts, it is necessary to be able to store the photons in the circuit while the offline preparation involved in probabalistic gates such as the CZ-gate take place. In this case, we need to be concerned with losses that may occur during storage. One method of evaluating circuit integrity is the fidelity Fqm[1]:
q√ √ 2 Fqm = T r( ρinρout ρin)
Where ρ is the density matrix. Memory can fail not just through loss, but through decoherence or mode-mismatching in the input and output modes. Mode-mismatching compromises photon indistinguishability in a circuit [6], and is related to both the detectors resolution and to the integral overlap of photon wavefunctions. The resolution can be thought of as spectral uncertainty, meaning the detector may be unable to distinguish between spectral components. Quantum memories need to be able to produce high fidelity photons, and to couple photons into the memory with a low probability of failure.
VII. A PROPOSED IMPROVEMENT ON KLM: CLUSTER STATES
FIG. 4. Cluster-state representation of a Bell state[1]
A cluster-state is a highly-entangled state of multiple qubits [7]. Graphically, physical qubits are represented√ by nodes, whose connections represent entanglement. Each node is a qubit prepared in the state |+i = 1/ 2(|0i + |1i). Then CZ gates are applied between qubits whose vertices are connected. Essentially, the CZ gate is used to add quibits to a cluster chain. This will actually be a two-step process as the CZ-gate is, as discussed, necessarily probabilistic for photonic qubits. The process will consist of a probabilistic offline preparation, in which the disconnected qubit is teleported. After this has succeeded, we then teleport the qubit at the end of the chain.
For quantum computing, we require both vertical and horizontal connections, as a linear chain cluster-state can be simulated on a classical computer [7].
If the cluster state is arranged into a 2-D lattice, the quantum computation essentially consists of single-qubit mea- surements on a column of qubits, whose outcomes determine the basis of measurement for the next column. Larger clusters can be constructed by combining smaller ”microclusters” [7]. 8
FIG. 5. A simple example of a two-dimensional cluster state [1]
We can use cluster-states to simulate circuit-based LOQM by performing single-qubit transformations and mea- surements on a cluster state. However, these operations are irreversible, as they destroy the qubits of the cluster state. As a result, measurement outcomes need to be fed forward through the cluster so that they may have an effect on later operations. This means that the cluster needs to be growing as it is being measured. An advantage of cluster-state quantum computing is a reduction in the number of necessary resources compared to the KLM protocol [7]. This is a interesting (if complicated!) direction for quantum computing with photonic qubits.
VIII. CONCLUSION
This paper discusses some of the very basics of quantum computing with photonic qubits, linear optics, and prob- abalistic two-qubit gates. The KLM protocol showed that despite the fact that photons do not directly interact, it is possible to create a scalable architecture for quantum computing using linear optics. The difficulty was resolved by way of an induced interaction using projective measurements, taking advantage of quantum teleportation in order to perform probabilistic measurements offline before. Since KLM, there have been many exciting and promising developments and proposals for reducing the number of resources, correcting for errors, and further refining the scheme of using photonic qubits and linear optics for quantum computing.
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