Exchange-Free Computation on an Unknown Qubit at a Distance

The generalisation of counterfactual communication---where classical information is sent without exchanging particles, or exchange-free---to transporting quantum states (counterportation) poses significant experimental challenges. Here, we show how to directly communicate an arbitrary qubit, not only exchange-free but also without the sender having to implement a quantum object locally, paving the way for a much more feasible future demonstration. More remarkably, we propose an exchange-free protocol that allows one party to directly enact, by means of a suitable program, any computation on a remote second party's unknown qubit. Further, we show how the first party can in principle directly enact any desired quantum algorithm, such as Shor's, on a remote second party's programmable quantum circuit, that is without any particles travelling between them.

Quantum physics opens up the surprising possibility of obtaining knowledge from, or through, places where no information-carrying particles have been. This was first proposed and subsequently demonstrated experimentally in the context of computing [1,2], where the result of a computation is learnt based on the phenomena of interaction-free measurement and the Zeno effect [3][4][5][6][7]. More specifically, without any photons entering or leaving an optical circuit, the result of a computation is obtained without the computer ever 'running'.
Just as intriguing was the proposal and subsequent experimental demonstration of a simple quantum scheme for allowing two remote parties to share a cryptographic random bit-string, without exchanging any informationcarrying particles [8,9]. The fact that the protocol had limited maximum-efficiency was not a serious a drawback for its purpose since the shared information was random, meaning failed attempts could simply be discarded in the end. This, however, begged the question whether efficient, deterministic communication was possible exchange-free, that is without particles crossing the communication channel.
In 2013, building on the ideas above, Salih et al devised a scheme allowing two remote parties to efficiently and deterministically share a message exchange-free, in the limit of a large number of protocol cycles and ideal practical implementation [10]. The protocol was recently demonstrated experimentally by Pan and colleagues [11]. Importantly, the previously-heated debate over whether the laws of physics even allow such communication (for both bit values) seems to be settling; Nature does allow exchange-free communication (and therefore computation) [12][13][14][15][16][17][18].
We present in what follows a protocol allowing a remote Bob to prepare any qubit he wishes at Alice, exchange-free. This is different from counterportation [19,20] in that Bob does not need to prepare a quantum object at his end (as a quantum superposition of blocking the optical communication channel and not blocking it) thus making the scheme much easier to implement. More generally, Bob can directly apply any arbitrary Bloch-sphere rotation to an unknown qubit at Alicein other words, any single-qubit quantum computation. Note that we use "exchange-free" and "counterfactual" interchangeably. While we describe an optical realisation using polarisation, the scheme is in principle applicable to any physical implementation.
Our protocol consists of a number of nested outer interferometers, each containing a number of inner interferometers, as in Salih et al's 2013 protocol [10]. We combine these interferometers into a device that we call a Phase Unit, allowing Bob to apply a relative phase to Alice's photonic qubit (Fig.1). We pair two Phase Units such that one applies some phase to Alice's H-polarised component, while the other applies an equal but oppositesign phase to her V -polarised component, resulting in â R z (θ) rotator. By chaining three suchR z (θ) rotators, interspersed with appropriate wave-plates, Bob can apply any arbitrary unitary to Alice's qubit, exchange-free (Fig.2).
Note, we define the Bloch sphere for polarisation such that the poles are |H and |V , and the rotations arê  [10], but with an added phase-module in the dashed box. The optical switches each alter the paths at different times in the protocol to allow the photon to do the correct number of cycles. Optical switch M 1 inserts the photon into the device, and keeps it in for M outer cycles; optical switches M 2 and M 3 cycle the photon around for N inner cycles per outer cycle. The Polarising Beamsplitters transmit H-polarised light, and reflect V -polarised light. The half wave plates are tuned to implementRy(θ) rotations on polarisation with θ of π, π/M and π/N , as shown in the figure.
As explained in the text, detectors DA and DB not clicking ensure that the photon has not been to Bob. After M outer cycles, the photon is sent by M 1 to the right. The photon only exits the Phase Unit if its polarisation had been flipped to V as a result of Bob blocking the channel (which he does by switching his Switchable Mirror on) because of the action of the Polarising Beamsplitter in the dashed box. Bob doesn't block for k runs (out of a maximum L), then blocks, allowing him to set the final phase of the photon, kπ/L, anywhere from 0 to π, in increments of π/L. An initially V -polarised photon can be put through an altered version of this device to add a phase to it (identical, except for the π/2 half-wave plate being moved to above M 1). The unit rotates Alice's qubit bŷ Rz(kπ/L).
for dummy variable θ, and Pauli matricesσ x,y,z . We first go through Salih et al's 2013 protocol. However, we describe the protocol, following [20], without any reference to either interaction-free measurement or the Zeno effect of [3,6]. In order to do this, we think of our detectors as being placed far enough, such that they perform no measurement before the photon had had time to exit the protocol. Any photonic component travelling towards either detector can thus be thought of as enter-ing a loss mode, meaning that if the photon exits the protocol successfully then it cannot have taken the path towards that detector, and the detector will subsequently not register a click.
To start with, a photon of state a |H +b |V enters the outer interferometer through a half wave plate (HWP) tuned to apply aR y (π/M ) rotation. The photon then enters a polarising beam splitter (PBS), which transmits horizontal polarisation, but reflects vertical polarisation.
The V -polarised component circles through a series of N inner interferometers, where, in each, it goes through a HWP tuned to apply aR y (π/N ) rotation, then through another PBS. The H-polarised component from this PBS passes across the channel, from Alice to Bob, who can choose to block or not block, by switching on or off his switchable mirror. If he blocks, this H-polarised component goes into a loss mode towards detector D B ; if not, it returns to Alice's side, recombines at another PBS with the V -polarised component, then enters the next inner interferometer. After the chain of N inner interferometers, the resulting components are then passed through one final PBS, sending any H-polarised component that has been to Bob into a loss mode towards detector D A , before being recombined at another PBS with the Hpolarised component from the arm of the outer interferometer. Importantly, neither detector clicking, ensures that the photon has not been to Bob.
As each inner interferometer appliesR y (π/N ), if Bob doesn't block, the rotations sum tô Therefore, the state after the inner interferometer chain is This means the V -polarised component becomes Hpolarised, entering the loss mode towards detector D A after the final PBS, meaning the only component of the wavefunction exiting the outer interferometer is the Hpolarised one that went via the outer arm.

Similarly, if Bob blocks for all inner interferometers,
Therefore, the state after an outer interferometer is meaning some V -polarised component exits the outer interferometer. If Bob, doesn't block, the outer cycle applies If he does block, the outer cycle applies We repeat this M times, starting with a H-polarised photon, and using a final PBS to split it into H-and V -polarised components.
As Alice applies aR y (π/M ) rotation at the start of each outer interferometer, if Bob doesn't block, the state of the photon after M outer cycles is Therefore, if the photon isn't lost, it remains Hpolarised. However, if Bob blocks, the photon after M outer cycles (as N → ∞) becomes V -polarised.
To prepare any qubit at Alice, Bob needs to apply a relative phase between Alice's two component, which can be represented as aR z (θ) rotation. Bob can implement this exchange-free using the device in Fig.1, for an Hpolarised component, relative to some other V -polarised component (e.g. one separated beforehand using a polarising beamsplitter).
We put this H-polarised component through one run of Salih et al's 2013 protocol, with Bob either always blocking or not blocking his channel. If he blocks, and the component exits V -polarised, Alice flips it to H-polarised and kicks it out of the device; however, if it is H-polarised, it gains a phase (π/L), and re-enters the device for another run.
This is repeated L times, with Bob blocking or not blocking for all outer cycles in a given run. After each run, the component goes into a PBS: if it is H-polarised, it gains a phase of π/L; if V -polarised, it is flipped to Hpolarised and sent out from the unit. Bob first doesn't block for k runs, applying a phase of kπ/L, then blocks, applying the transformation The overall protocol, incorporating multiple Phase Units from Fig.1, as well as polarising beamsplitters (which transmit horizontally-polarised, and reflect verticallypolarised, light), as well as a quarter wave plate and its adjoint (conjugate-transpose). The setup allows Bob to implement any arbitrary unitary on any initial pure state |ψ Alice inserts, entirely exchange-free.
When N is finite, the rotations applied by each outer cycle when Bob blocks are not complete, meaning one run (M outer cycles) doesn't fully rotate the state from H to V . However, given Bob only blocks after the component has had a phase applied to it, to kick the component out of the device, any erroneous H-polarised component can be kept in the device by Bob not blocking for the remaining L − k full runs afterwards, letting us treat the erroneous H-component as loss.
While coarse-grained for finite L, as L goes to infinity (with 0 ≤ k/L ≤ 1), Bob can generate any relative phase for Alice's qubit, from 0 to π. Further, by moving the π/2 half-wave plate from its location in Fig.1 to the input, a similar phase can be added to a V -polarised component, relative to a H-polarised component.
Moreover, the Phase Unit can be constructed to include Aharonov and Vaidman's clever modification of Salih et al's 2013 protocol [17], satisfying their weakmeasurement criterion for exchange-free communication. We do this by running the inner cycles for 2N cycles rather than N , except that for the case of Bob not blocking, he instead blocks for one of the 2N inner cycles, namely the N th inner cycle. This has the effect of helping to remove any lingering V component exiting the inner interferometer of Fig.1 due to imperfections in practical implementation.
We now use our Phase Unit as the building block for a protocol where Bob can implement any arbitrary unitary onto Alice's qubit, exchange-free. Any arbitrary 2 × 2 unitary matrix can be written aŝ U = e i(2α −β σz−γ σy−δ σz)/2 = e iα R z (β )R y (γ )R z (δ ) (10) Note, the factor of e iα can be ignored, as it provides global rather than relative phase, which is unphysical for a quantum state [21].
We can apply theR z (θ) rotations using the Phase Unit, and make aR y (θ) rotation by sandwiching aR z (θ) rotation between a −π/4-aligned Quarter Wave Plate, U QW P , and its adjoint,Û † QW P , wherê We set β = 2πβ/L, γ = 2πγ/L, δ = 2πδ/L (12) where, for the three Phase Unit runs, k is β, γ and δ. The Phase Units form components of the overall protocol, as shown in Fig.2. Here, Alice first splits her input state |ψ into H-and V -polarised components with a polarising beamsplitter (PBS), before putting each component through a Phase Unit, to generate equal and opposite phases on each. She recombines these at another PBS. Afterwards, she puts the components through a quarter wave plate, then through another run of PBS, Phase Unit, and PBS, then through the conjugatetranspose of the quarter-wave plate, tuned to convert the partialR z rotation (phase rotation) into a partialR y rotation. Finally, she applies another run of PBS, Phase Unit, and PBS to implement a secondR z rotation.
Using theseR z andR y rotations, Bob can implement any arbitrary rotation on the surface of the Bloch sphere on Alice's state. This can be used either to allow Bob to prepare an arbitrary pure state at Alice (if she inserts a known state, such as |H ), or to perform any arbitrary unitary transformation on Alice's qubit, without Bob necessarily knowing that input state.
Because the Phase Units output their respective photon components after Bob blocks for a run, the timing of which depends on the phase Bob wants to apply, there is a time-binning (a grouping of exit times into discrete bins) of the components from each Phase Unit correlated with the phase Bob applies in that unit. Bob can, on his side, compensate for the time-binning (given he knows the phase he is applying). Further, in order to locate the photon in time, Alice can detect the time of exit using a non-demolition single photon detector.
Alternatively, we could add a final pair of Phase Units with the value of k set to 3L − β − γ − δ (where L is the value of L for each of the first three Phase Unit pairs, and β, γ and δ are their respective k-values), but without phase plates (see Fig.1). This means that while no phase is applied, a time delay is still added to the components, meaning the photon always exits the overall device at a time proportional to 3L, rather than β, γ and/or δ as before. This makes the time of exit uncorrelated to Bob's unitary, which means Alice can know in advance the expected exit time of her photon from the protocol (without needing to perform a non-demolition measurement to find it). When considering a finite number of outer and inner cycles, there is a nonzero probability of the photon not returning to Alice, which reduces the protocol's efficiency. The survival probability of a photon going through a Phase Unit is plotted in Fig.3. The survival probability for the overall protocol is the product of the survival probability for the three Phase Units: As expected, as {M, N } → ∞, the survival probability goes to one. Regardless, postselection renormalises Alice's output state such that if Alice receives an output photon, it will be in a pure state. Thus, for our set-up, given ideal optical components, the rotation enacted on Alice's qubit is always the rotation Bob has applied, not just for any L, but also for any N , M , and k.
Interestingly, a Phase Unit, which outputs a photon into one of L different time bins depending on the number of runs Bob blocks, could be adapted to sending, FIG. 4. Quantum circuit diagram, showing how a 3-qubit gate applying a controlled-controlled unitary U can be constructed from two-qubit gates along with single-qubit gates, where U is some unitary transformation, and V 2 = U [22]. Using our exchange-free single-qubit gate, a classical Bob can directly simulate the control action on Alice's photonic qubits. Since any quantum circuit can be constructed using 2-qubit gates along with single-qubit ones, our exchange-free single-qubit gate allows Bob in principle to directly program any quantum algorithm at Alice, without exchanging any photons.
exchange-free, a classical logical state of dimension d greater than two-a "dit", rather than a bit. We leave the details to an upcoming paper.
We now show how our exchange-free protocol enabling arbitrary single-qubit operations, can in principle allow a classical Bob to directly enact any quantum algorithm he wishes on Alice's qubits, without exchanging any particles with her. This is based on the fact any quantum algorithm can be efficiently constructed from 2-qubit operations (such as CNOT) and single-qubit ones. Our protocol already enables exchange-free single-qubit operations, i.e. gates. Thus, if Bob can directly activate or not, a 2-qubit gate at Alice, exchange-free, then directly programming an entire quantum algorithm at Alice using these two building blocks becomes possible. The quantum network of figure Fig.4 shows how a 3-qubit controlled-controlled gate, applying some unitary U to the target qubit at Alice, can be constructed from 2-qubit controlled gates [22]. (Some controlled-controlled gates can have more implementable circuits than the general one given here [21].) A classical Bob, at the top end of Fig.4, uses our exchange-free single-qubit gate to simulate the control action on Alice's photonic qubits. For simulating the CNOT gate, he can choose to either apply the identity transformation, representing control-bit |0 , or apply an X transformation, representing controlbit |1 . For the controlled-V gate, he can choose to either apply the identity transformation, again representing control-bit |0 , or apply a V transformation, representing control-bit |1 . In this scenario, we envisage an optical programmable circuit, with exchange-free singlequbit gates acting on Alice's qubits that Bob can directly program, and 2-qubit gates acting on Alice's qubits that he can directly choose to activate, exchange-free.
In summary, we have presented a protocol allowing Bob to directly perform any computation on a remote Alice's qubit, without exchanging any photons between them. We use this to show how, in principle, Bob can directly enact any quantum algorithm at Alice, exchangefree.
We thank Claudio Marinelli for helping to bring this collaboration together. This work was supported by the Engineering and Physical Sciences Research Council (Grants EP/P510269/1, EP/T001011/1 and EP/L024020/1). a much simpler protocol for Bob to prepare exchange-free a qubit with real, positive probability amplitudes. This is also based on Salih et al's 2013 counterfactual communication protocol [10]. However, here, Bob doesn't block for the first M − k outer cycles, then blocks for the rest. In order to eliminate the error resulting from a finite number of blocked inner cycles, Alice introduces loss, attenuating the outer arm of the interferometer on her side by a factor of cos (π/2N ) N for each outer cycle. This means, before the final PBS, the state is By postselecting on Alice's photon successfully exiting the protocol, she receives the state By choosing a ratio of k/M , this gives any real, positive qubit superposition (for infinite M ), forming an arc parameterised by k/M between the two poles of the Poincaré sphere. This means Bob is directly applying a form ofR y rotation (albeit one requiring an input state of |H , to provide a fullR y rotation, we would need a way to combine any arbitrary pair of orthogonal polarisation states, analogously to a PBS).

Kraus Operator Notation
Viewing exchange-free communication more abstractly, we consider the communication channel in Kraus operator notation.
Here, we associate a channel X to the set of Kraus operators {X i } i which describe its action on a given density operator such that In general, for channels X and Y, their composition can be written, and we denote the N -fold composition of a channel X N (ρ) := X • X • · · · • X (ρ). In this manner we can define three channels in this protocol: first that constituting Bob's action on the channel, b, that goes via him, when he blocks/doesn't block Each cycle of Alice's inner-interferometer is given by y,(a1,b) , and imposing that the initial state in Bob's mode is vacuum, and omitting it from the output state by tracing it out, we have that over the N inner cycles one finds channels on Alice's mode a1 given by, where any channel acting on a larger Hilbert space than that on which it's defined acts as identity channel, i.e. B N B ∼ B N B ⊗ 1. We find when Bob blocks/doesn't block: then finally the effect this has overall as the channel created by a chain of M outer interferometers on Alice's inner and outer interferometer (V and H) modes, when Bob blocks/doesn't block: Therefore, we find A B 12 ∼ c 1 |1 a2 0 a1 1 a2 0 a1 | , c 2 |0 a2 1 a1 0 a2 1 a1 | , c 3 |0 a2 1 a1 1 a2 0 a1 | , where coefficients c 1 ,c 2 , c 3 and c 4 are functions of M and N , with c 2 and c 3 going to 1, and c 1 and c 4 going to zero, as N and M go to infinity. This means one run (of M outer cycles of N inner cycles each) acts as a perfect optical switch in this limit, turning H to V (and vice-versa) if Bob blocks, and implementing identity if he doesn't.