Jan 3, 2008 - Avalanche photodiode (APD): A device for counting photons that ab- sorbs a single ... Beam splitter: A linear optical device that partially transmits, and par- tially reflects, an ..... This may not sound very useful if you are in fact ...
Quantum Computing Using Optics. G.J. Milburn and A.G.White Centre for Quantum Computer Technology, The University of Queensland, QLD 4072, Australia. January 3, 2008
Glossary. Avalanche photodiode (APD): A device for counting photons that absorbs a single photon and, with some probability, produces a single electrical signal. Beam splitter: A linear optical device that partially transmits, and partially reflects, an incoming beam of light. Bell inequality: The results of local measurements of dichotomic observables on each component of two correlated classical systems satisfy a correlation function that is less than or equal to a universal bound. This bound can be exceeded by correlated quantum systems. Bell states: Four orthogonal, maximally entangled states of two qubits that violate a Bell inequality. Cluster State:A highly entangled state of many qubits that enables quantum computation by sequences of conditional single qubit measurements, each conditioned on the results of previous measurements. c-sign: A two-qubit gate that leaves all logical states unchanged unless both qubits take the value one, in which case the state suffers a π phase shift. Entangled state: The state of a multi-component quantum system which cannot be expressed as a convex combination of tensor product states of each subsystem. Entangled states cannot be prepared by local operations on each subsystem, even when supplemented by classical communication. HOM interference: Hong, Ou and Mandel discovered that when indistinguishable single photon pulses arrive simultaneously at each of the two input 1
ports of a fifty-fifty beam splitter, the probability for detecting two photons, co-incidently, at each of the two output ports drops to zero. In other words, the photons are either both reflected or both transmitted. Mixed State: A quantum state that is not completely known and thus has non zero-entropy. Photon: A single quantum excitation of an orthonormal optical mode. A field in such a state has definite intensity and completely random phase, so that the average field amplitude is zero. Pure state: A quantum state that is completely known and thus has zero entropy. Quantum computation: The ability to process information in a physical device using unitary evolution of superpositions of the physical states that encode the logical states. Qubit: The fundamental unit of quantum information in which two orthogonal states encode one bit of information. Unlike a classical bit, the physical system that forms the qubit can be in a superposition of the two logical states simultaneously. Quantum teleportation: A quantum communication protocol based on measurement, feed-forward and shared entanglement. Shor’s algorithm:An algorithm for finding the prime factors of large integers by unitarily processing information in a quantum computer. Tomography: A measurement scheme for experimentally determining a quantum state in which a large sequence of physical systems, prepared in the same state, are subjected to measurements of a carefully chosen set of physical observables. Unitary: A transformation of a quantum state that is physically, and thus logically, reversible. Unitary transformations take pure states to pure quantum states.
1
Definition.
Quantum computation is a new approach to information processing based on physical devices that operate according to the quantum principles of superposition and unitary evolution[1, 2]. This enables more efficient algorithms than are available for a computer operating according to classical principles, the most significant of which is Shor’s efficient algorithm for finding the fac2
tors of a large prime integer[3]. There is no known efficient algorithm for this task on conventional computing hardware. Optical implementations of quantum computing have largely focussed on encoding quantum information using single photon states of light. For example, a single photon could be excited to one of two carefully defined orthogonal mode functions of the field with different momentum directions. However, as optical photons do not interact with each other directly, physical devices that enable one encoded bit of information to unitarily change another are hard to implement. In principle it can be done using a Kerr nonlinearity as was noted long ago[4, 5], but Kerr nonlinear phase shifts are too small that to be useful. Knill et al.[6] discovered another way in which the state of one photon could be made to act conditionally on the state of another using a measurement based scheme. We discuss this approach in some detail here as it has led to experiments that have already demonstrated many of the key elements required for quantum computation with optics.
2
Introduction.
The optical fibre communication network is the largest artificial complex system on the planet, enabling the internet, the growth of economies and a massive connectivity of minds presaging a but dimly seen revolution. Information, encoded in pulses of light, courses through the system at more than 10 terabits per second. It is hard to believe that it is barely 20 years since the first transoceanic optical fibre was installed, yet the system is still growing at an astonishing pace. The insatiable appetite for bandwidth in modern economies does not look like abating any time soon. The entire system is held together by thin fibres of glass guiding pulses of light. The huge increase in bandwidth that modern communication networks have enabled follows directly from the very high carrier frequency of optical pulses. A wealth of modulation techniques have been developed to exploit the potential of this bandwidth: it is currently limited only by the speed of the switches in the network required to control the flow of information. Early networks required the switches to be largely electronic and this meant first converting the light pulses to electronic pulses. However increasingly these switches are being replaced with all-optical devices. The routers and switches in the optical fibre network are essentially small computers processing packets of information at the fastest possible speeds. 3
In the 1980’s there was a research program to build computers entirely from optical switches processing information encoded optically. The astonishing progress in silicon technology meant that optical computers could never compete with the shrinking scale of semiconductors. The size of an optical switch is largely determined by the wavelength of light. This is beginning to change with the rise of plasmonics and nano photonics. However this constraint does not pose a problem for an optical communication system that spans the entire planet and much of the early work on all-optical switches found its way into the optical fibre system. Optical fibre networks operate by the principles of classical physics; the computers they connect operate by the principles of classical logic. The semiconductors used as light sources and detectors function by quantum principles, but these do not influence network operation or logic. This is the physics and logic of Newton, Faraday and Maxwell. The motion of charges in semiconductor circuits is governed by the classical understanding of electric and magnetic fields, while the light pulses coursing down a glass fibre are perfectly well described by Maxwell’s theory of electromagnetic radiation. However this situation will inevitably change and the first indications are already upon us. We have known since the early part of the last century that the deepest description of the physical world is not classical but quantum. It is now clear that the quantum world enables new computation and communication tasks that are difficult, if not impossible, in a classical world. Over the same period that the optical communication system has been built, a small group of visionaries have speculated on the ultimate limits to conventional computation. There are two strands to this question. The first strand takes us down a road of ever decreasing dimensions. The astonishing growth in semiconductor technology is the direct result of acquiring the technical ability to make transistors smaller and smaller so that billions can fit onto a single chip. A very natural question is: how small can it get? Ultimately the answer to this question is the domain of quantum mechanics. The second strand joins a more abstract path that began with von Neumann and Turing, and took a quantum twist in the mid 1980s when Feynman [1] asked if a physical computer operating by quantum principles would be a more efficient computational machine than a conventional classical computer. We now know that the answer is yes. The quantum description of light began with Einstein in 1905 with his explanation of the photoelectric effect [7]:
4
when a light ray starting from a point is propagated, the energy is not continuously distributed over an ever increasing volume, but it consists of a finite number of energy quanta, localized in space, which move without being divided and which can be absorbed or emitted only as a whole. In Einstein’s explanation, the energy of each quanta, or photon to use the modern word, is proportional to its frequency while the intensity of the light determines the number of photons per second passing through some given area. Einstein’s insight is now routinely confirmed in a semiconductor device known as the avalanche photodiode (APD). This is a device that produces a current pulse when a single photon is absorbed. If we turn the intensity of the optical source down to very low levels and connect the APD to an audio amplifier we can hear the individual clicks as the photons arrive on the surface of the detector. In this operational sense, a single photon is a detection event, figure 1
classical signal-in source detector
classical signal-out
Figure 1: A source generates a sequence of optical pulses conditional on some classical input, eg electrical pulses. A detector registers a sequence of classical electrical pulses that we interpret as due to the propagation of individual photons from source to detector. We might call this the ballistic picture of a photon. The weak pulses of light used in conventional optical communication systems contain a huge number of photons, and furthermore, the number of photons per pulse is not fixed but fluctuates from pulse to pulse. This is a direct consequence of the kind of light source that is used, the laser. The laser is a true quantum device but it necessarily produces pulses of light with an indeterminate number of photons. The reason for this is intimately connected with the coherence properties of laser light. In a laser 5
light source both the amplitude of the light and its phase fluctuate as little as quantum theory will permit. However these fluctuations are related by an uncertainty principle: if we try and produce a sequence of pulses with a well defined number of photons we necessarily make the phase random from pulse to pulse (we shall make this idea more precise in what follows). The random phase would destroy the key feature of first order optical coherence that makes a laser so useful. Are there any light sources that controllably produce pulses of light each with one and only one photon per pulse? Until very recently, the answer was no. However the discovery of optical communication schemes (quantum key distribution) and computation schemes (as described below) based on quantum principles have given a strong incentive for building such sources. We will now explain how single photons enable quantum computation and postpone our discussion of exactly what a single photon source is to later.
3
Quantum computation with single photons.
Let us now consider the experiment depicted in figure 2. A source (labelled D for downward) is producing a sequence of single photons which are directed towards a 50/50 beam splitter. This is a classical optical device which in the wave theory wold be described as simply dividing the wave intensity equally between a reflected beam and a transmitted beam. If we had a source of D le
ng
si
U ot
ph on pu e
ls
U D
Figure 2: A single photon pulse incident on a beam splitter in the downward (D) direction. It can be detected in the upward going direction (U) or the downward going direction (D). 6
very high intensity, each of the photodetectors in the reflected path and the transmitted path would record on average an equal number of counts per second. As we reduce the intensity to the single photon level however, we see an irreducible uncertainty in which detector will register the photon in each trial. On average it is still the case that over many trials, one half are recorded at the D-detector and one half at the U-detector. If we were to persist in our ballistic view of a single photon, we might be tempted to say that each photon is either reflected or transmitted at random; it is a simple coin toss. This picture would adequately capture the experimental facts, for this experiment. The beam splitter experiment has a binary outcome: the photon is either detected at U or D. To encode the result at the output we need a single bit of information. Indeed if we allow another single photon source into the upward going input to the beam splitter, we will need a single binary number to encode the input state as well. The kind of encoding we have just described is called dual rail. In more precise terms we have encoded a single bit into one of two perfectly distinguishable momentum states of a single photon. At first sight, it would appear that the experiment we have just described is a fanciful coin toss. That this is not the case can be seen if we ‘toss the coin again‘, that is to say, we take the photon after the beam splitter and, instead of running it into a photodetector, we reflect it back onto an identical beam splitter and then ask for the probability for it to be reflected or transmitted (see figure 3). It is well known that in this case we can adjust D le
ng
si
U=1 ot
ph on e
ls
pu
U D=0
Figure 3: A single photon pulse incident on a beam splitter in the downward (D) direction. It can be detected in the upward going direction (U) or the downward going direction (D). things so that the probability for the photon to be detected at U is certain. Irreducible uncertainty has ben turned into certainty simply by changing the experimental conditions for detecting the photon, not by changing the state 7
of the single photon source. Note that the price we pay for certainty is complete loss of knowledge of what happened to the photon at the left beam splitter. A photon can be detected at either detector in figure 3 in two indistinguishable ways corresponding to the two (unknowable) outcomes of reflection or transmission at the left beam splitter. Returning to the simple experiment in figure 1, can we say that it is a simple coin toss before the detectors register the photon? Does the output of the beam splitter really encode one bit of information? No. We capture the dual quantum/classical nature of the state of a single photon after a beam splitter by saying that it is a superposition of the two mutually exclusive alternatives. In such a case the output is said to encode a single quantum bit, or qubit. Quantum computation and communication runs on qubits not bits. The quantum description of the state of the photon after a single beam splitter requires us to give a list of the probability amplitudes for the two mutually exclusive possibilities for detection in either the downward or the upward direction. The detection probabilities are then given by the square of the probability amplitudes. For example the input state in figure 1 is certain to be in the downward direction, so the ordered pair of probability amplitudes (1, 0) in which the first (second) entry corresponds to detection in the D (U) direction. After a general (not 50/50) beam splitter the state of the photon is depicted by the ordered pair (r, t) where |r|2+|t|2 =1. So in fact the beam splitter has enacted the linear transformation (1, 0) → (r, t). One might think that the input state (0, 1) is transformed in exactly the same way. However these two input states are physically distinguishable: in one case the photon is going down and in the other it is going up with certainty. Mathematically this is captured by the fact that the input vectors are orthogonal. The beam splitter is a perfectly reversible device and does not destroy information by making two distinguishable alternatives indistinguishable.The problem is avoided when we note that a full quantum theory of beam splitters will in fact show that (0, 1) → (−r, t). There is a central lesson here: operations on qubits must not take two physically distinguishable states and make them indistinguishable. We call this sort of transformation unitary. We now make a change to more conventional notation. The input states (1, 0) and (0, 1) are written as |1, 0i and |0, 1i (ordering is preserved) so that
8
the beam splitter transformations are then written as |1, 0i → r|1, 0i + t|0, 1i |0, 1i → −r|1, 0i + t|0, 1i
(1) (2)
We now formalise the concept of a qubit by making a distinction between the logical state of a qubit and the physical states of the system used to encode it. In the example used here the relation ship is |0i = |1, 0i |1i = |0, 1i
(3) (4)
We then say that we have a qubit code that uses one photon and two optical modes per qubit. The beam splitter transformation on the logical code acts like |0i → r|0i + t|1i |1i → −r|0i + t|1i
(5) (6)
This transformation on logical qubits is called a one-qubit gate. If we want to encode two qubits in this dual rail scheme we will need two single photon pulses and four modes, which may be achieved by two independent beam splitters. In a notation that keeps track of the ordering of the beam splitters by an ordering of states from left to right, the logical state |1, 0i would then be represented physically by |1i ⊗ |0i = |0, 1i ⊗ |1, 0i
(7)
Note that the number of mutually distinguishable output states from N beam splitters (and N photons) increases exponentially as 2N . This simply means that for the logical code of N qubits there are 2N possible logical states. If we continue in this fashion we are not going to have a very compact notation. We can make things easier by using the fact that our qubits are ordered corresponding to a physical order of the underlying beam splitters. We can then represent a state, for example |1i ⊗ |0i ⊗ |1i ⊗ |0i, by regarding it as the binary code for an integer, in this case |10i as 10 = 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20 . It is then clear that with N physical resources we can code 2N binary numbers (or their corresponding integers). If we could somehow act on all 9
these numbers simultaneously we might have a very efficient processor. This is precisely the hope of quantum computation. For example, if we pass N photons downward through N independent beam splitters, the state at the output is superposition state |0i → 2−N/2
N −1 2X
x=0
|xi ,
(8)
an equal superposition of all integers from zero up to 2N −1. If we could somehow continue to postpone measurement and act unitarily on all these numbers simultaneously, we might gain a considerable advantage in computational efficiency. This was the idea first captured in precise form by David Deutsch [2]. Suppose for example we set aside some number, say N, of photons to encode the input register and some number, say K, to encode an output register. We first prepare the N photons so as to encode the state produced in Eq. (8) and the K to encode the state |0i. Then the unitary processing through some, as yet unspecified, physical device could implement the transformation X X |xi|0i → |xi|f (x)i (9) x
x
for some function f (we drop the normalisation for simplicity). Of course now when we read out the output register we only get one value of the function with a random distribution. This may not sound very useful if you are in fact interested in a particular value of the function, but in that case why do all computations of the function at once? The primary reason why you might want to evaluate a function on all inputs is when you are not so much interested in any particular value but rather some global property of the function itself — for example is it constant? It is precisely that kind of computation that Deutsch [8] showed could be done more efficiently on a quantum computer. A far more interesting example was described by Shor [3] who gave an efficient quantum algorithm for finding the period of a modular function. The particular modular function he used is part of an algorithm for finding the prime factors of large integers, so that in effect he gave an efficient quantum algorithm for finding such prime factors. The assumed intractability of this problem for a classical computer is why it is used as a method for public-key encryption. If someone had a quantum computer all these crypto systems 10
would be vulnerable to attack. We describe below a simple optical experiment which captures the kind of methods by which a quantum computer could implement Shor’s algorithm. Of course no one yet has a quantum computer capable of posing a threat to public-key crypto systems, although it begins to look like it might be possible. The problem is to figure out what kind of physical systems would enable the arrow in Eq.(9). Various suggestions have been made including single photon optical systems, which we discuss below. At the level of logical qubits we need two types of elementary transformations. The first we have already seen: It is the single qubit transformation in Eq.(6). In addition all we need is some form of interaction that correlates the state of at least two qubits. One example is the controlled-sign or c-sign, gate, |xi|yi → (−1)x·y |xi|yi
(10)
For single photon, dual rail encoding this presents a major problem. If we look at the c-sign gate at the level of physical qubits it will require some kind of intensity dependent phase change so that only a two photon component acquires a π phase change. This is possible with a Kerr nonlinearity as was noted long ago [4, 5]. Nonlinear optical materials with an intensity dependent refractive index do indeed exist and are generically referred to as Kerr nonlinear materials. The problem is that Kerr nonlinear phase shifts are so small that to expect an intensity as low as two photons to give a π phase change in an optical material of practical size is to expect too much. Fortunately there is another way.
4
Conditional optical two-qubit gates.
Measurements play quite a different role in quantum mechanics than they do in classical mechanics. In the latter, it is the objective of accurate measurement to revel the pre-existing values of dynamical variables. Of course in reality both the measurement and the preparation of the initial state are subject to noise. However it is assumed that both may be sufficiently refined to reveal the true dynamical state of a system. If we know all there is to know about a classical system, all measurements are dispersion free in principle. Quantum mechanics is an irreducibly statistical theory. This means that even if a system has been prepared in state about which we have maximal knowledge, there will be at least one measurement the results of which are 11
uncertain. Fortunately there is also at least one physical quantity that may be measured for which the results are certain. If we make a perfect measurement of this quantity we have in effect prepared the system in a pure state of which we have maximal knowledge. Measurement thus plays an essential role in state preparation, where knowledge of the prepared state is heralded by the measurement result. Can we use the unique role that measurement plays in quantum mechanics to conditionally implement a required state transformation? Consider the situation shown in figure 4. In a dual rail scenario, two modes are mixed on a beam splitter. One mode is assumed to be in the vacuum state (a) or a one photon state (b), while the other mode is arbitrary. A single photon counter is placed in the output port of mode-2. What is the conditional state of mode-1 given a count of n photons? |ψ 1
2
|0
> >
1
|ψ
(0)
>
|ψ
'0' 2
>
'1'
1
2
2
1
|1
>
|ψ
(1)
>
case b
case a
Figure 4: A conditional state transformation conditioned on photon counting measurements. The conditional state of mode a1 is given by (unnormalised), ˆ |ψ˜(i) i1 = Υ(i)|ψi 1
where
ˆ Υ(i) = 2 hi|U(θ)|ii2
(11) (12)
with i = 1, 0. Here U(θ) is the unitary transformation implemented by the beam splitter as we described in the introduction. The probability for the specified photo count event is given by, ˆ P (i) =1 hψ|Υ† (i)Υ(i)|ψi (13) 1, which fixes the normalisation of the state, 1 |ψ˜(i) i1 . |ψ (i) i1 = p P (i) 12
(14)
It is then possible to show that ∞ X (cos θ − 1)n † n n ˆ (a1 ) a1 Υ(0) = n! n=0 ˆ ˆ (0) − sin2 θa†1 Υ ˆ (0) a1 . Υ(1) = cos Υ
In order to see how we can use these kind of transformations to effect a csign gate, consider the situation shown in figure 5. Three optical modes are
|ψ>
a0
π
|ψ'> θ2
|1>
a1 θ1
|0>
θ3
a2
n2 = 1
n3= 0
Figure 5: A conditional state transformation on three optical modes, conditioned on photon counting measurements on the ancilla modes a2 , a3 . The signal mode, a1 is subjected to a π phase shift. mixed on a sequence of three beam splitters with beam splitter parameters θi , with corresponding reflection and transmission amplitudes, ri = cos θi , ti = sin θi . The ancilla modes, a1 , a2 are restricted to be in the single photon states |1i2 , |0i3 respectively. We will assume that the signal mode, a0 , is restricted to have at most two photons, thus |ψi = α|0i0 + β|1i0 + γ|2i0 (15) The reason we are only interested in two photon states is that in dual rail encoding a general two qubit state can have at most two photons. We can now chose the beam splitter parameters so that, conditional on the detectors both registering a one, the signal state is transformed as |ψi → |ψ ′ i = α|0i + β|1i − γ|2i
(16)
with a probability that is independent of the input state |ψi. This last condition is essential as in a quantum computation, the input state to a general two qubit gate is completely unknown. We will call this transformation the ns (for nonlinear sign shift) gate. This can be achieved using [6]: 13
θ1 = −θ3 = 22.5 deg and θ2 = 65.53 deg. The probability of the conditioning event (n2 = 1, n3 = 0) is 1/4. Note that we can’t be sure in a given trial if the correct transformation will be implemented. Such a gate is called a nondeterministic gate. However the key point is that success is heralded by the results on the photon counters (assuming ideal operation). We can now proceed to a c-sign gate in the dual rail basis. Consider the situation depicted in figure 6. We first take two dual rail qubits encoding
{
Q1 in
Q2
{
0
1 0
NS1
1 0
1
1
in
1 0
NS2
0
1 0
}
Q1out
}
Q2out
Figure 6: A conditional state transformation conditioned on photon counting measurements. A c-sign gate that works with probability of 1/16. It uses HOM interference and two ns gates. for |1i|1i. The single photon components of each qubit are directed towards a 50/50 beam splitter where they overlap perfectly in space and time and produces a state of the form |0i2 |2i3 + |2i2 |0i3,a effect known as Hong-OuMandel (HOM) interference [9]. We then insert an ns gate into each output arm of the HOM interference. When the conditional gates in each arm work, which occurs with probability 1/16, the state is multiplied by an overall minus sign. Finally we direct these modes towards another HOM interference. The output state is thus seen to be −|1i|1i. One easily checks the three other cases for the input logical states to see that this device implements the c-sign gate with a probability of 1/16 and successful operation is heralded. Clearly a sequence of nondeterministic gates is not going to be much use: the probability of success after a few steps will be exponentially small. The key idea in using nondeterministic gates for quantum computation is based on the idea of gate teleportation of Gottesmann and Chuang [10]. In quantum teleportation an unknown quantum state can be transferred from A to B provided A and B first share an entangled state. Gottesmann and Chuang realised that it is possible to simultaneously teleport a two qubit quantum state and implement a two qubit gate in the process by first applying the 14
gate to the entangled state that A and B share prior to teleportation. We use a non deterministic ns gate to prepare the required entangled state, and only complete the teleportation when the this stage is known to work. The teleportation step itself is non deterministic but, as we see below, by using the appropriate entangled resource the teleportation step can be made near deterministic. The near deterministic teleportation protocol requires only photon counting and fast feed-forward. We do not need to make measurements in a Bell basis. A nondeterministic teleportation measurement is shown in figure 7. The client state is a one photon state in mode-0 α|0i0 + β|1i0 and we prepare the entangled ancilla state |t1 i = |01i12 + |10i12 (17) where mode-1 is held by the sender, A, and mode-2 is held by the receiver, B. For simplicity we omit normalisation constants wherever possible. This an ancilla state is easily generated from |01i12 by means of a beam splitter.
k 50/50
|1>
l k+l=1
50/50
|0> state preparation |1>|0>+|0>|1>
Figure 7: A partial teleportation system for single photons states using a linear optics. If the total count is n0 + n1 = 0 or n0 + n1 = 2, an effective measurement has been made on the client state and the teleportation has failed. However if n0 + n1 = 1, which occurs with probability 0.5, teleportation succeeds with
15
the two possible conditional states being α|0i2 + β|1i2 if n0 = 1, n1 = 0 α|0i2 − β|1i2 if n0 = 0, n1 = 1
(18) (19)
When successful, this procedure implements a joint measurement on modes 0 and 1. In the conventional deterministic teleportation protocol the joint measurement is a simultaneous measurement of the commuting operators XX and ZZ where X, Z are respectively the Pauli-x and Pauli-z operators. This is a Bell measurement. In the teleportation protocol considered here, we only have a partial Bell measurement. We will refer to it as a nondeterministic teleportation protocol, T1/2 . Note that teleportation failure is detected and corresponds to a photon number measurement of the state of the client qubit. Detected number measurements are a very special kind of error and can be easily corrected by a suitable error correction protocol. For further details see [6] and the review [11] The next step is to use T1/2 to effect a conditional sign flip c-sign1/4 which succeeds with probability 1/4. Note that to implement c-sign on two bosonic qubits in modes 1, 2 and 3, 4 respectively, we can first teleport the first modes of each qubit to two new modes (labeled 6 and 8) and then apply c-sign to the new modes. When using T1/2 , we may need to apply a sign correction. Since this commutes with c-sign, there is nothing preventing us from applying c-sign to the prepared state before performing the measurements. The implementation is shown in Fig. 8 and now consists of first trying to prepare two copies of |t1 i with c-sign already applied, and then performing two partial Bell measurements. Given the prepared state, the probability of success is (1/2)2 . The state can be prepared using c-sign1/16 , which means that the preparation has to be retried an average of 16 times before it is possible to proceed. The probability of successful teleportation can be boosted to 1−1/(n+1) using more entangled resource state of the kind |tn i1...2n =
n X j=0
|1ij |0in−j |0ij |1in−j .
(20)
The notation |aij means |ai|ai . . ., j times. The modes are labeled from 1 to 2n, left to right. Note that the state exists in the space of n bosonic qubits, where the k’th qubit is encoded in modes n + k and k (in this order). 16
Figure 8: A c-sign two qubit gate with teleportation to increase success probability to 1/4. When using the basic teleportation protocol (T1), we may need to apply a sign correction. Since this commutes with c-sign, it is possible to apply c-sign to the prepared state before performing the measurements, reducing the implementation of c-sign to a state-preparation (outlined) and two teleportations. The two teleportation measurements each succeed with probability 1/2, giving a net success probability of 1/4. The correction operations C1 consist of applying the phase shifter when required by the measurement outcomes.
17
We can teleport the state α|0i0 + α|1i0 using |tn i1...2n . We first couple the client mode to half of the ancilla modes by applying an n + 1 point Fourier transform on modes 0 to n. This is defined by the mode transformation n
X 1 ak → √ ω kl al n + 1 l=0
(21)
where ω = ei2π/(n+1) . This transformation does not change the total photon number and is implementable with passive linear optics. After applying the Fourier transform, we measure the number of photons in each of the modes 0 to n. If the measurement detects k bosons altogether, it is possible to show [6] that if 0 < k < n + 1, then the teleported state appears in mode n + k and only needs to be corrected by applying a phase shift. The modes 2n−l are in state 1 for 0 ≤ l < (n − k) and can be reused in future preparations requiring single bosons. The modes are in state 0 for n−k < l < n. If k = 0 or k = n+1 an effective measurement of the client is made, and the teleportation fails. The probability of these two events is 1/(n+ 1), regardless of the input. Note that again failure is detected and corresponds to measurements in the basis |0i, |1i with the outcome known. Note that both the necessary correction and the receiving mode are unknown until after the measurement.
5
Cluster state methods.
About the same time it was realised that measurements would provide a path to optical single photon computing, Raussendorf and Briegel [12] gave an independent and remarkably novel method by which measurement alone could be used to do quantum information processing. In their approach to quantum computation, an array of qubits is initially prepared in a special entangled state called a cluster state. The computation then proceeds by making a sequence of single qubit measurements. Each measurement is made in a basis that depends on prior measurement outcomes; in other words, the results of past measurements are fed forward to determine the basis for future measurements. Subsequently Popescu showed that the linear optical measurement based scheme of Knill et al. can be interpreted as a Raussendorf and Briegel measurement based quantum computation[13]. Nielsen [14] realised that the LOQC model of [6] could be used to efficiently assemble the cluster using the nondeterministic teleportation tn . As 18
we saw the failure mode of this gate constituted an accidental measurement of the qubit in the computational basis. The key point is that such an error does not destroy the entire assembled cluster but merely detaches one qubit from the cluster. This enables a protocol to be devised that produces a cluster that grows on average. The LOQC cluster state method dramatically reduces the number of optical elements required to implement the original LOQC scheme. Of course if large single photon Kerr nonlinearities were available, the optical cluster state method could be made deterministic [15]. In its simplest form the Raussendorf and Breigel scheme begins with a two dimensional array of qubits. Each qubit is prepared in a superposition of the computational basis states, |0i + |1i. Then an entangling operation is performed between nearest neighbour qubits in the lattice using the twoqubit controlled sign operation |xi|yi 7→ (−1)xy |xi|yi
(22)
In the next step one or more qubits are measured in a particular basis and depending on the results of that measurement another basis is chosen for subsequent qubit measurements. Any circuit model of a quantum algorithm can be mapped onto the two dimensional lattice through a sequence of conditional measurements on subsets of qubits. The key difficulty in doing this with a dual rail optical scheme is that, as we have noted, the transformation in Eq.(22) is very difficult to implement using a deterministic unitary transformation as optical nonlinearities are too small. However a conditional scheme of the kind discussed above might work by conditionally entangling sequences of qubits, provided failure at any point did not destroy the entire developing lattice of entanglement. This is indeed the case because of a key feature of the LOQC model of [6] scheme. If a teleportation gate failure is heralded, it corresponds to an effective measurement of one of the qubits at input. This feature was used by Knill et al. to establish the scalability of the scheme as detected measurements errors can easily be protected by a suitable code. The importance of this feature for a conditional cluster state assembly is that an accidental measurement of one of the qubits in a developing cluster simply detaches it from the cluster without destroying the remaining entanglement. The picture then emerges of a kind of random cluster assembly in which the cluster grows when a gate succeeds and gets pruned if a gate fails. As long as the probability for gate success is greater than 0.5 there is 19
a chance that the cluster overall will grow. As this does not require a very large teleportation resource,|tn i1...2n , it can be achieved with quite modest overheads of linear optics and single photons. For this reason optical cluster state methods are preferred over the original LOQC scheme of Knill et al. A number of schemes have been proposed to efficiently assemble a cluster via this probabilistic growth. A recent application of percolation theory is a good example of the kind of optimisation that is required [16].
6 6.1
Experimental Implementations. Two qubit gates.
Considerable progress has been made on demonstrating conditional two-qubit gates with single photon states. After the initial proposal was made by Knill, Laflamme and Milburn [6], there was a flurry of theoretical work to propose linear-optical two-qubit gates that would be easier to realise experimentally. These can be divided into two categories, internal-ancilla gates [17, 18, 19, 20], where the ancilla photons are intrinsic to the operation of the logic gates, and external-ancilla gates [21, 22], where ancilla photons can be used to verify correct gate operation by performing a quantum non-demolition measurement [23] on the gate outputs. There are a variety of configurations of internal-ancilla gates, including simplified [17] and efficient [18] versions of the KLM gate; and gates that gain in efficiency using entangled ancilla photons [19, 20]. An entangled internal-ancilla gate was soon realised in Johns Hopkins, where entangling operation was suggested by a 61.5±7.4% visibility fringe [24]. An unambiguous demonstration of entangling gate operation was performed at the University of Queensland with an external-ancilla gate [25]: all four entangled Bell states were produced as a function of only the logical values of the input qubits, for a single operating condition of the gate. Both these gates filtered on photon-number, i.e. they required the four or two input photons to be detected to signal successful gate operation. An important requirement for LOQC is that it must be possible to detect successful gate operation by measurement of the ancilla photons and then feed-forward this information to the logic photons: this was realised with an external-ancilla gate at the University of Vienna [26]. Linear-optic gates, both internal and external ancilla, have achieved a 20
wide range of firsts and proof-of-principle demonstrations: the first full characterisation of a quantum-logic gate, in any architecture [27]; production of cluster [28] and graph [29] states, and their use for Grover’s algorithm [28]; production of the highest entanglement [30] and fastest gate operation [31] of any physical architecture; and the first demonstration of Shor’s algorithm exhibiting the core processes, coherent control, and resultant entangled states required in a full-scale implementation [32, 33]. discard 1/3
{ {
Cin
Tin
1/3 1/2
1/2 1/3
} }
Cout
Tout
discard
Figure 9: An external-ancilla cnot gate. When the control is in the logicalone state, the control and target photons interfere non-classically at the central 1/3 beam splitter which causes a π phase-shift in the upper arm of the central interferometer and the target state qubit is flipped. The qubit value of the control is unchanged. Correct operation has probability 1/9 and occurs when a single photon is detected in each output; this can be done with quantum non-demolition (QND) measurements using external ancilla photons [23] or by filtering on photon number in the final detection. Experimentally, the two modes of each qubit are distinguished by orthogonal polarisations. These may be split into separate spatial modes and sent through normal beamsplitters [25, 27] or left in one spatial mode and sent through partially-polarising beamsplitters [30, 34, 35]. Measuring gate operation is non-trivial. Further, it is desirable to diagnose, and if possible correct, error behaviours introduced by a real gate, such as phase or bit-flip errors, which can induce the wrong amount or type 21
of entanglement, or decoherence. To date, there have been a wide variety of measures used to gauge the quality of two-qubit gates. A comprehensive comparison of the various measures, and an architecture-independent measurement standard for two-qubit gates, is given in [36]. The key is quantum process tomography, which allows reconstruction of the quantum state transfer function of the gate. Tomography requires sampling the statistics of a fixed number of measurement outcomes, at least 256 for a two-qubit gate. With these statistics data inversion can be devised to reconstruct the gate process. From this we can calculate its overlap, or fidelity, with respect to the ideal quantum-logic gate, e.g. in recent experiments at the University of Queensland we obtained a process fidelity of Fp =98.2±0.3% for a controlledπ/4 gate [37]. The process fidelity is itself not a metric [38], but forms the basis of several, such as the average gate fidelity, i.e. the fidelity of every gate output state with the ideal, averaged over every possible input state. This is given simply by, dFp +1 , (23) F= d+1 where d is the dimension of the process matrix, e.g. d=16 for two-qubit gates. These measures are useful for comparing gate performances, but do not provide an error probability-per-gate to allow direct comparison with faulttolerance thresholds. A recent publication introduces a semidefinite programming technique to do exactly this, using the measured process matrix [39]. Such fault-tolerant benchmarking identifies the magnitude, but not the source of errors—critical in identifying the critical technology path for improving gate operation. This requires a comprehensive theoretical model of the quantum-logic gate, and its errors. Using such a model, Ref. [39] identifies small amounts of multi-photon emission as the dominant, yet previously unrecognised, source of error in linear-optical quantum computing—higherorder terms of 0.3−0.8% lead to gate errors of ∼16%! Removing multi-photon emission puts photonic quantum computing within striking distance of a recently predicted fault-tolerance threshold [40].
6.2
Quantum optical algorithms.
The simplest useful algorithm for quantum optical information processing is the quantum repeater. This is more conventionally referred to as quantum communication protocol rather than an algorithm, but realistic systems will require few qubit quantum computers to do entanglement distillation and 22
purification giving it an algorithmic flavour. Furthermore quantum repeaters could enable distributed quantum computation. We also mention it here as the various quantum optical repeater protocols that have been suggested [41, 42, 43] will have immediate application in long distance quantum key distribution and thus are pivotal to developing quantum optical networks across the global optical fibre network. This raises many new questions in the area of communication complexity and a great deal of work remains to be done to understand such systems [44, 45]. The general idea of a quantum repeater is sketched in figure 10. There are three key elements at each node of the network: (i) a pair of entangled qubits, (ii) quantum memories, (iii) a few qubit quantum computer for purification distillation. In the first step two entangled pairs are produced at the origin, 1. memory joint measurement
memory
2.
3.
4. joint measurement
5
6
7 quantum processor
quantum processor
8.
Figure 10: A simple schematic for a quantum repeater. one of each pair is held there, while the other member of each pair is sent 23
to distant locations. A measurement is made on the two qubits kept at the origin at step 1, leaving partially entangled qubits at the remote locations, step 2. The remote qubits are then loaded into quantum memory, step 3. The process is repeated, steps 4 and 5 until two qubits are held at the remote locations. A purification algorithm is then run at each remote location so that finally a maximally entangled pair is held at the remote locations separated by twice the distance that separated the pairs in step 1. In the quantum optical realisation both the generation of entanglement and purification can be done by heralded non deterministic processes. The previous scheme of course requires a good quantum memory and a great deal of research is underway to develop such systems for photons. One promising approach is to use polarised atomic ensembles [46]. The quantum memory itself may need to have error correction algorithms running to maintain the coherence of the entangled pairs until they required for some future quantum information processing task. Many current cryptographic protocols rely on the computational difficulty of finding the prime factors of a large number: a small increase in the size of the number leads to an exponential increase in computational resources. Shor’s quantum algorithm for factoring composite numbers faces no such limitation [3], and its realization represents a major challenge in quantum computation. Only one step of Shor’s algorithm to find the factors of a number N requires a quantum routine. Given a randomly chosen co-prime C (where 1 1. Then we can approximate the positive frequency components by 1/2 r X ∞ c ~Ωn (+) E (x, t) = i an e−iωn (t−x/c) (27) 2ǫ0 Ac L n=0 where A is a characteristic transverse area. This operator has dimensions of electric field. In order to simplify the dimensions we now define a field operator that has dimensions of s−1/2 . Taking the continuum limit we thus define the positive frequency operator Z ∞ ′ −iΩ(t−x/c) 1 √ dω ′a(ω ′ )e−iω (t−x/c) (28) a(x, t) = e 2π −∞ where we have made a change of variable ω 7→ Ω + ω ′ and used the fact that Ω >> 1 to set the lower limit of integration to minus infinity, and [a(ω1 ), a† (ω2 )] = δ(ω1 − ω2 )
(29)
In this form the moment n(x, t) = ha† (x, t)a(x, t)i has units of s−1 . This moment determines the probability per unit time (the count rate) to count a photon at space-time point (x, t) [52]. The field operators a(t) and a† (t) can be taken to describe the field emitted from the end of an optical cavity, which selects the directionality. We will contrast single photon states with multimode coherent states defined by a multimode displacement operator acting on the vacuum D|0i, defined implicitly by D † a(ω)D = a(ω) + α(ω) (30) where consistent with proceeding assumptions, α(ω) is peaked at ω = 0 which corresponds to a carrier frequency Ω >> 1. The average field amplitude for this state is Z ∞ −iΩ(t−x/c) 1 √ ha(x, t)i = e α(ω)e−iω(t−x/c) ≡ α(x, t)e−iΩ(t−x/c) (31) 2π −∞ which implicitly defines the average complex amplitude of the field as the Fourier transform of the frequency dependent displacements α(ω) We can 27
also calculate the probability per unit time to detect a photon in this state at space-time point (x, t). This is given by n(x, t) = |α(x, t)|2 . Note that in this case the second order moment ha† (x, t)a(x, t)i factories, a result characteristic of fields with first order coherence. A coherent state is is closest to our intuitive idea of a classical electromagnetic field. The multimode single photon state is defined by Z ∞ |1i = ν(ω)a† (ω)|0i (32) −∞
Normalisation requires that Z
∞
dω|ν(ω)|2 = 1
(33)
−∞
This last condition implies that the total number of photons, integrated over all modes, is unity, Z ∞
dωha† (ω)a(ω)i = 1
(34)
−∞
This state has zero average field amplitude but n(x, t) = |ν(t − x/c)|2
(35)
where ν(t) is the Fourier transform of ν(ω). So while the state has zero average field amplitude there is apparently some sense in which the coherence implicit in the superposition of Eq.(32) is manifest. In fact comparing this to the case of a coherent state, Eq(31), we see that the expression for n(t) is also determined by the Fourier transform of a coherent amplitude. For this state the function ν(φ) is periodic in the phase φ = t − x/c and it is not difficult to choose a form with a well defined pulse sequence. However care should be exercised in interpreting these pulses. They do not represent a sequence of pulses each with one photon rather they represent a single photon coherently superposed over all pulses. Once a photon is counted in a particular pulse, the field is returned to the vacuum state. A review of current efforts to produce such a state may be found in [53]. More recent results may be found in [54, 55, 56, 57]. In the absence of true single photon pulse sources, most of the experimental work we have described has been done with a conditional source based on photon pair production in spontaneous parametric down conversion. We now give a model of the kind of state such sources produce. 28
In the case of a continuous wave pump, the entangled two photon produced by SPDC may be well approximated by Z ∞ Z ∞ |ψi = dω1 dω2 ρ(ω1 , ω2 )a† (ω1 )b† (ω2 )|0i , (36) 0
0
where a, b designate distinguishable modes, for example, distinguished either by polarisation or wave vector [58]. More precisely, this is the conditional state given that a down conversion process has in fact taken place at all. The probability per unit time for this event is the down conversion efficiency. For spontaneous parametric down conversion with a continuous pump field at frequency 2Ω we can approximate ρ(ω1 , ω2 ) = δ(ω1 + ω2 − 2Ω)α(ω1 )
(37)
We now make the change of variable ǫ = Ω − ω1 and assume that the bandwidth, B, over which α(ω1 ) is significantly different from zero is such that Ω >> B, then we can write Z ∞ |ψi = dǫβ(ǫ)a† (−ǫ)b† (ǫ)|0i (38) −∞
where we have defined β(ǫ) = α(Ω − ǫ) and a(−ǫ) ≡ a(Ω − ǫ), b(ǫ) ≡ b(Ω + ǫ). We will also assume that β(−ǫ) = β(ǫ). Normalisation of |ψi requires that Z ∞ |β(ǫ)|2 = 1 (39) −∞
The probability per unit time to detect a photon from this field with a unit efficiency detector is in fact unity, na (t) = ha† (t)a(t)i = 1 The probability per unit time to detect a photon from mode-a is thus independent of time. This simply means that photons will be counted at randomly distributed times from each of the fields a, b. This is a reflection of the fact that the state Eq(38) is invariant under time translations in a Lorentzian frame. On the other hand let us now compute the coincidence rate, C(t, t′ ) = hψ|a† (x, t′ )a(x, t′ )b† (x, t)b(x, t)|ψi
(40)
This is given by Z C(t, t ) =
∞
′
2 −iǫ(t′ −t)
β(ǫ)e
−∞
˜ )|2 = |β(τ ≡ C(τ ) 29
(41) (42) (43)
where τ = t′ − t. The coincidence rate is thus symmetrical about t′ − t = 0 and is peaked at t′ = t. Even though photons are detected at random, independently from each beam, they are highly correlated in time. In the case of spontaneous SPDC the distributions function β(ǫ) is given approximately [59] 1 β(ǫ) ∝ 2 (44) κ + ω2 We can now consider a heralded single photon source made by detecting one of the photon pairs and then ask for the kind of single photon state conditionally produced in the other mode. In the case of a CW pump, we first must provide a temporal filter on the detected mode. One can think of this as a time dependent detector that is switched on an off over some time interval. In frequency domain this is simply a filter. If such a detector is placed at the b mode, the conditional state of the a mode is given by Eq.(32) with 2 2 ν(ω) ∝ e−ω /κ (45) which corresponds to a Gaussian temporal pulse. If we drive SPDC with a pulsed pump, the pump itself provides a natural temporal filter. In this case the frequency distribution function in Eq.(37) is not delta correlated but takes the form [58] ρ(ω1 , ω2 ) = exp −(ω1 + ω2 − 2Ω)2 /σp2 α(ω1 ) (46) where Ω is the pump carrier frequency and σp is the bandwidth of the pump pulse. It is still the case however that no photon down conversion event may take place within the pump pulse window. Thus the source is not a deterministic single photon source. Migdall [60] has proposed a way to over come this by multiplexing many heralded conditional SPDC sources with a conditioning detection on one mode of each of the multiplexed pairs. Another approach has been implemented by the Polzik group [61]. They used a cavity to enhance the parametric down conversion to implement a frequency tunable source of heralded single photons with a narrow bandwidth of 8 MHz. This approach is particularly important as frequency tunability makes the source compatible with atomic quantum memories.
30
8
Future Directions.
Optical systems are certain to be used for future quantum communication protocols. Indeed the first steps have already been taken with quantum key distribution. In this article we have seen that it is also possible to process quantum information optically using heralded non deterministic schemes of various kinds and simple examples have been implemented experimentally. This greatly enhances the practicability of quantum communication schemes that require some quantum processing, such as quantum repeaters. While this has not yet been realised in practice, we expect that the first demonstrations are not far away. In the effort to produce single photon sources we are learning new ways to encode and process information in optical pulses. We have seen that coherent communication can be done using single photons despite the fact that the average field amplitude for such states is zero. If information can be encoded and decoded on photon number states, this would represent a major step beyond quantum communication protocols like quantum key distribution. It is far from clear however if optical schemes will be viable for large scale quantum computation. Currently ion trap schemes and schemes based on super-conducting devices offer the most likely way forward for quantum computation per se. However a number of investigators are turning to the concept of a hybrid quantum computer which combines optical and matter based qubits. Optical qubits with heralded non deterministic processing combined with matter based quantum memories is poised to make significant achievements. Hybrid schemes offer a path to distributed quantum computation between many nodes each made up of a few hundred qubits. The nodes do not need to be far apart: they could simply be different parts of a single quantum computation device. However, if you will permit us some license, it is not too difficult to imagine a single quantum computation spanning the entire planet, with matter based nodes connected by quantum optical communication channels. Such a system would hold in its web massively entangled quantum states and exhibit a complexity that would make our current optical communication system look rather simple by comparison.
31
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