EExperimental replication of single-qubit quantum phase gates
M. Miˇcuda, R. St´arek, I. Straka, M. Mikov´a, M. Sedl´ak, M. Jeˇzek, and J. Fiur´aˇsek
Department of Optics, Palack´y University, 17. listopadu 1192/12, CZ-771 46 Olomouc, Czech Republic
We experimentally demonstrate the underlying physical mechanism of the recently proposed pro-tocol for superreplication of quantum phase gates [W. D¨ur, P. Sekatski, and M. Skotiniotis, Phys.Rev. Lett. , 120503 (2015)], which allows to produce up to N high-fidelity replicas from N input copies in the limit of large N . Our implementation of 1 → PACS numbers: 03.67.-a, 42.50.Ex, 03.65.Ud, 03.67.Hk
I. INTRODUCTION
Laws of quantum mechanics impose fundamental lim-its on our ability to replicate quantum information [1, 2].The quantum no-cloning theorem for instance ensuresthat quantum correlations cannot be exploited for super-luminal signaling [2], and it guarantees the security ofquantum key distribution [3]. These fundamental andpractical impacts of the no-cloning theorem stimulated awide range of studies on approximate quantum cloningand numerous optimal quantum cloning machines for var-ious sets of input states were designed and experimen-tally demonstrated [4]. Recently, the concept of quan-tum cloning was extended from states to quantum op-erations [5, 6], which brings in new interesting aspectsand features. In particular, it was shown very recentlythat starting from N copies of a single-qubit unitary op-eration U one can deterministically generate up to N copies of this operation with an exponentially small er-ror [7, 8]. As pointed out in Ref. [9], the observed N limit of the number of achievable almost perfect clonesis deeply connected to the Heisenberg limit on quantumphase estimation.In this paper we experimentally demonstrate the un-derlying physical mechanism of the superreplication ofquantum gates, see Fig. 1. Specifically, we consider 1 → U ( φ ) = | (cid:105)(cid:104) | + e iφ | (cid:105)(cid:104) | , φ ∈ [0 , π ) . (1)As shown in Fig. 1(b), we impose suitable quantum cor-relations between the two signal qubits and a third aux-iliary qubit with the help of a three-qubit quantum Tof-foli gate, we apply the operation U ( φ ) to the auxiliaryqubit, and we finally erase the correlations between theauxiliary qubit and the signal qubits. This latter partof the protocol can be in principle accomplished unitar-ily by a second application of the Toffoli gate, see Fig.1(b), but we instead choose to erase the correlations bymeans of a suitable measurement on the auxiliary qubit,see Fig. 1(c), which is much less demanding and makes the protocol experimentally feasible. Remarkably, thisgate-replication scheme also simultaneously acts as a de-vice which adds a control to an arbitrary unknown single-qubit phase gate and converts it to a two-qubit controlledphase gate [10–12].We experimentally implement the gate replication pro-tocol using a linear optical setup, where qubits are en-coded into states of single photons. We perform fullquantum process tomography of the replicated quantumgates and characterize the performance of the replicationprotocol by quantum gate fidelity. The linear opticalquantum Toffoli gate which we utilize [13] is probabilis-tic and operates in the coincidence basis, hence requirespostselection, similarly as other linear optical quantumgates [14]. Although the deterministic nature of the gatesuperreplication protocol is one of its important features,our proof-of-principle probabilistic experiment still al-lows us to successfully demonstrate the underlying phys-ical mechanism of quantum gate replication. (a) (b)(c) (d) | ψ i| i U ⊗ N V V † | ψ i| i U | ψ i| i U W U
FIG. 1: (Color online) (a) Quantum circuit for superreplica-tion of single-qubit quantum phase gates U ( φ ) [7]. (b) 1 → U ( φ )into a two-qubit controlled-phase gate. For details, see text. a r X i v : . [ qu a n t - ph ] J a n II. GATE SUPERREPLICATION PROTOCOL
Let us first briefly review the generic N → M super-replication protocol for phase gates U ( φ ) [7]. We intro-duce the following notation for M -qubit computationalbasis states, | m (cid:105) = | m m . . . m M (cid:105) , where m ∈ [0 , M − ]and m i ∈ { , } represent digits of binary representationof m . It holds that U ( φ ) ⊗ M | m (cid:105) = e i | m | φ | m (cid:105) , (2)where | m | = (cid:80) Mi =1 m i denotes the Hamming weight of an M -bit binary string m m . . . m M . All basis states | m (cid:105) with the same Hamming weight | m | acquire the samephase shift | m | φ and the number of such states C M | m | obeys binomial distribution, C M | m | = (cid:0) M | m | (cid:1) . This degeneracy is exploited in the superreplicationscheme illustrated in Fig. 1(a) [7]. The whole protocolcan be divided into three steps. First, a unitary operation V imprints information about the Hamming weight of the M -qubit signal state of system A onto an auxiliary N -qubit system B , which is initially prepared in state | (cid:105) B .In the joint computational basis | m (cid:105) A | n (cid:105) B we explicitlyhave V | m (cid:105) A | (cid:105) B = | m (cid:105) A | (cid:105) B , | m | < m min ,V | m (cid:105) A | (cid:105) B = | m (cid:105) A | k ( m ) (cid:105) B , m min ≤ | m | < m max ,V | m (cid:105) A | (cid:105) B = | m (cid:105) A | N − (cid:105) B , | m | ≥ m max . (3)Here each | k ( m ) (cid:105) is an N -qubit computational basisstate chosen such that its Hamming weight satisfies | k | = | m | − m min , and the lower and upper thresholds onthe Hamming weight are chosen symmetrically about thevalue M/ C M | m | is maximized, m min = (cid:100) M − N (cid:101) , m max = (cid:100) M + N (cid:101) .In the second step of the protocol, the N replicatedgates U ( φ ) ⊗ N are applied to the N auxiliary qubits. Fi-nally, in the third step, the system A is disentangled fromthe auxiliary qubits by applying an inverse unitary op-eration V − . This sequence of operations results in thefollowing unitary transformation on the M -qubit systemA, | m (cid:105) → | m (cid:105) | m | < m min , | m (cid:105) → e i ( | m |− m min ) φ | m (cid:105) m min ≤ | m | < m max , | m (cid:105) → e iNφ | m (cid:105) | m | ≥ m max . (4)According to the Choi-Jamiolkowski isomorphism [15,16], an M -qubit unitary operation U can be representedby a pure maximally entangled state of 2 M qubits, | Φ U (cid:105) = I ⊗ U | Φ (cid:105) = 12 M/ M − (cid:88) m =0 | m (cid:105) ⊗ U | m (cid:105) , (5)where | Φ (cid:105) = 2 − M/ (cid:80) M − m =0 | m (cid:105)| m (cid:105) . Similarity of twounitary operations U and U can be quantified by the fidelity of the corresponding Choi-Jamiolkowski states, F U U = |(cid:104) Φ U | Φ U (cid:105)| , which can be expressed as F U U = | Tr[ U † U ] | / M . It can be shown that in thelimit of large N the fidelity of transformation (4) withthe ideal operation U ( φ ) ⊗ M is exponentially close to onewhen M = N − α , α >
0. Consequently, N copies of U ( φ ) suffice to faithfully implement up to N copies of U ( φ ). An intuitive explanation for this result is that inthe asymptotic limit the function C M | m | becomes Gaus-sian with a width that scales as √ M . If M < N ,then the Hamming weights of an exponentially largemajority of the basis states are located in the interval m min ≤ | m | < m max , and the superrepliction protocolinduces correct relative phase shifts for all such states.Although experimental implementation of the super-replication protocol for large M is beyond the reach ofcurrent technology, the main mechanism of the super-replication can be demonstrated already for N = 1 and M = 2. In this case we have m min = 1 and m max = 2,and it follows from Eq. (3) that V is the three-qubit quan-tum Toffoli gate [11, 13, 17–21], which flips the state ofthe auxiliary qubit if and only if both signal qubits arein state | (cid:105) , see Fig. 1(b). Assuming a pure input state | ψ in (cid:105) A = c | (cid:105) + c | (cid:105) + c | (cid:105) + c | (cid:105) of the signalqubits and recalling that the auxiliary qubit is initiallyin state | (cid:105) , the three-qubit state after the application ofthe Toffoli gate reads | Ψ (cid:105) AB = c | (cid:105)| (cid:105) + c | (cid:105)| (cid:105) + c | (cid:105)| (cid:105) + c | (cid:105)| (cid:105) . (6)If we now apply the unitary operation U ( φ ) to the auxil-iary qubit, and then disentangle this qubit from the restby another application of the Toffoli gate, we obtain apure output state of the two signal qubits, | ψ out (cid:105) A = c | (cid:105) + c | (cid:105) + c | (cid:105) + e iφ c | (cid:105) . (7)The resulting two-qubit unitary operation thus reads CU ( φ ) = | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + | (cid:105)(cid:104) | + e iφ | (cid:105)(cid:104) | . (8)The theoretical scheme in Fig. 1(b) can be further sim-plified by replacing the second Toffoli gate with measure-ment of the output auxiliary qubit in the superpositionbasis |±(cid:105) = √ ( | (cid:105) ± | (cid:105) ) followed by a feed-forward, seeFig. 1(c). In particular, a two-qubit controlled-Z gateshould be applied to the signal qubits if the auxiliaryqubit is projected onto state |−(cid:105) while no action is re-quired when it is projected onto state | + (cid:105) .Fidelity of transformation (8) with the two replicas ofthe phase gate U ( φ ) ⊗ U ( φ ) reads F UU ( φ ) = 18 (5 + 3 cos φ ) . (9)We note that the fidelity can be made phase independentand equal to a mean fidelity ¯ F UU = for all φ by twirling[22], which would make the replication procedure phase-covariant. To apply the twirling, one has to generate a | ψ i| i H H U ( φ ) FIG. 2: (Color online) Quantum circuit for optimal 1 → U ( φ ) [6]. uniformly distributed random phase shift θ and apply atransformation U ( θ ) to the ancilla qubit, in addition tothe replicated operation U ( φ ). Subsequently, an inverseunitary operation U † ( θ ) ⊗ U † ( θ ) is applied to the twooutput signal qubits. Since U ( φ ) U ( θ ) = U ( φ + θ ), itholds that for a fixed θ the fidelity of replication of U ( φ )is given by Eq. (9), where φ is replaced with φ + θ . Afteraveraging over the random uniformly distributed θ wefind that the fidelity becomes equal to the mean fidelityfor any φ .The mean fidelity ¯ F UU exceeds the fidelity which isachievable by applying the transformation U ( φ ) to oneof the qubits while no operation is applied to the secondqubit. Mean fidelity of 5 / U ( φ ) is applied to afixed probe state | + (cid:105) , optimal phase estimation is per-formed on the output U ( φ ) | + (cid:105) , and the estimated phase φ E determines the operation U ( φ E ) ⊗ U ( φ E ) applied tothe two signal qubits. However, this latter procedure con-ceptually differs from the generic superreplication proto-col and adds noise to the output state.Remarkably, Eq. (8) shows that the quantum circuitdepicted in Fig. 1(b) can be also interpreted as a schemefor deterministic conversion of an arbitrary single-qubitunitary phase gate U ( φ ) into a two-qubit controlled uni-tary gate CU ( φ ) [10–12]. While adding control to anarbitrary unknown single-qubit operation U is known tobe impossible [23], this task becomes feasible providedthat one of the eigenvalues and eigenstates of the uni-tary operation is known [10], which is precisely the case ofthe single-qubit phase gates (1) considered in the presentwork.An appealing feature of the superreplication protocolis that it is conceptually simple and universally applica-ble to any N and M . Nevertheless, for specific N and M one can further optimize this scheme to maximize thereplication fidelity. In particular, the optimal scheme for1 → et al. [6] and is depicted in Fig. 2.Average fidelity of this optimal cloning procedure reads F = (3 + 2 √ / ≈ . U ( φ ) toancilla qubit at their cores, but the optimal cloning alsorequires two additional controlled-Hadamard (CH) gateson the input qubits and two controlled-NOT gates onthe output qubits instead of a single Toffoli gate. Thelatter two CNOT gates can be replaced by a measure-ment of the ancilla qubit in the superposition basis |±(cid:105) followed by a suitable feed-forward on the signal qubits,similarly as in Fig. 1(c). However, the two CH gates areunavoidable, which makes the circuit in Fig. 2 much moredifficult to implement than the circuit in Fig. 1(c). III. EXPERIMENTAL SETUP
We experimentally implement the circuit in Fig. 1(c)which demonstrates the underlying physical mechanismof superreplication of quantum phase gates. The experi-mental setup is shown in Fig. 3 and its core is formed byour recently demonstrated linear optical quantum Toffoligate [13]. We utilize time-correlated photon pairs gen-erated in the process of spontaneous parametric down-conversion in a nonlinear crystal pumped by a laser diode.The two signal qubits are encoded into the spatial and po-larization degrees of freedom of the signal photon [24–27],respectively, while the auxiliary qubit is represented bypolarization state of the idler photon [13, 27]. The spatialqubit is supported by an inherently stable Mach-Zehnderinterferometer formed by two calcite beam-displacers BDwhich introduce a transversal spatial offset between verti-cally and horizontally polarized beam. Arbitrary productinput state of the three qubits can be prepared with theuse of quarter- and half-wave plates (QWP and HWP),and the output states can be measured in an arbitraryproduct basis with the help of a combination of wave-plates, polarizing beam splitters and single-photon de-tectors. The quantum Toffoli gate is implemented by atwo-photon interference on a partially polarizing beamsplitter PPBS that fully transmits horizontally polar-ized photons and partially reflects vertically polarizedphotons [13, 28–31]. Similarly to other linear optical FIG. 3: (Color online) Experimental setup. HWP - half-waveplate, QWP - quarter-wave plate, PPBS - partially polariz-ing beam splitter with reflectances R V = 2 / R H = 0for vertical and horizontal polarizations, respectively, PBS -polarizing beam splitter, BD - calcite beam displacer, APD- single-photon detector. Inset (a) shows the implementedquantum circuit and inset (b) depicts the single-photon de-tection block DB. FIG. 4: (Color online) Real and imaginary parts of the experimentally determined quantum process matrices χ representingthe two-qubit operations R implemented on the two signal qubits are plotted for four values of the phase shift: φ = 0 (a), φ = π/ φ = π (c), and φ = 3 π/ χ ) = 1. quantum gates [14], this Toffoli gate operates in the co-incidence basis [32] and its success is indicated by simul-taneous detection of a single photon at each output port.More details about the experimental setup can be foundin Refs. [13, 33].The phase gate U ( φ ) on the polarization state of theidler photon is implemented using a sequence of suitablyrotated quarter- and half-wave plates, see Fig. 3. At theoutput, the polarization of the idler photon is measuredin the superposition basis |±(cid:105) , and we accept only thoseevents where it is projected onto state | + (cid:105) . This ensuresthat we do not have to apply any feed-forward operationonto the signal qubits. While such conditioning reducesthe success rate of the protocol by a factor of 2, it doesnot represent a fundamental modification, because thelinear optical Toffoli gate utilized in our experiment isprobabilistic in any case. Since both signal qubits areencoded into a state of a single photon, a real-time elec-trooptical feed-forward scheme [34–37] could be in prin-ciple exploited to apply the controlled-Z gate to signalqubits if the auxiliary idler qubit is projected onto state |−(cid:105) . IV. EXPERIMENTAL RESULTS
We have performed a full tomographic characteriza-tion of the experimentally implemented quantum gatereplication protocol. The measurements were performedfor eight different values of the phase shift, φ = k π , k = 0 , , · · · ,
7, and for each φ we have reconstructedthe resulting quantum operation R on the two signalqubits. We make use of the Choi-Jamiolkowski iso-morphism [15, 16] and represent R by its correspond- ing quantum process matrix χ = I ⊗ R ( | Φ (cid:105)(cid:104) Φ | ), where | Φ (cid:105) = (cid:80) j,k =0 | jk (cid:105)| jk (cid:105) denotes a maximally entangledstate of four qubits, and I represents a two-qubit iden-tity operation, c.f. also Eq. (5). For each φ , the pro-cess matrix χ was reconstructed from the experimen-tal data using a Maximum Likelihood estimation [38].The experimentally determined matrices χ are plottedin Fig. 4 for four values of the phase shift φ : 0, π , π ,and π . Additional results are provided in the Appendixwhich for comparison also contains plots of the corre-sponding theoretical matrices χ th of the ideal two-qubitunitary operations (8). We quantify the performanceof our experimental scheme by the quantum process fi-delity F CU of each implemented operation with the cor-responding unitary operation CU ( φ ) that would be im-plemented by our setup under ideal experimental condi-tions. Recall that fidelity of a general completely positivemap χ with a unitary operation U [39, 40] is defined as F = (cid:104) Φ U | χ | Φ U (cid:105) / Tr[ χ ]. The fidelity F CU ( φ ) calculatedfrom the experimentally reconstructed process matrices χ is plotted in Fig. 5(a). We find that F CU is in therange of 0 . ≤ F CU ≤ .
897 and the mean fidelityreads ¯ F CU = 0 .
872 which indicates high-quality perfor-mance of our setup for all φ .The experimentally determined fidelities F CU are con-sistent with the fidelity of the Toffoli gate, F T = 0 . , which exhibits R V = 0 . R H = 0 .
017 instead of R V = 2 / R H = 0,limited visibility of two-photon interference V = 0 . FIG. 5: (Color online) Experimentally determined quantumgate fidelities F CU ( φ ) (a) and F UU ( φ ) (b) are plotted for theeight values of φ that were probed experimentally. The solidline in panel (b) represents the ideal theoretical dependence(9) and the dashed line is the least-square fit of the form A + B cos φ , with A = 0 .
543 and B = 0 . [33]. Additional factors that may influence the experi-ment include imperfections of the wave plates and polar-izing beam splitters that serve for state preparation andanalysis. A more detailed discussion of these effects andtheir influence on performance of the experimental setupcan be found in Ref. [33].We now turn our attention to characterization of gatereplication. For this purpose, we calculate the fidelity F UU ( φ ) of the implemented operations with U ( φ ) ⊗ U ( φ ).The results are plotted in Fig. 5(b) and we can see that F UU exhibits the expected periodic dependence on φ aspredicted by the theoretical formula (9). Due to variousimperfections, the observed fidelities are smaller than thetheoretical prediction, and the dashed line in Fig. 5(b)shows the least-square fit of the form A + B cos φ to thedata, which well describes the observed dependence of F UU ( φ ) on φ . The mean fidelity reads ¯ F UU = 0 . . U ( φ ) is applied to one qubit while no operationis performed on the other qubit.Statistical uncertainties of fidelities were estimated as-suming Poissonian statistics of the measured two-photoncoincidence counts. Since the fidelities are estimated in-directly from the reconstructed quantum process matri-ces χ , we have performed repeated Monte Carlo simula-tions of the experiment, and for each run of the simula-tion we have determined the quantum process matrix χ and the fidelities F UU and F CU . This yielded an ensem-ble of fidelities which was used to calculate the statistical errors. The maximum statistical uncertainty of fidelitythat we obtain reads 0.0013 (one standard deviation). V. SUMMARY
In summary, we have successfully experimentallydemonstrated the underlying physical mechanism of su-perreplication of quantum phase gates. Specifically, wehave imprinted information about the structure of theinput state of signal qubits onto an auxiliary qubit via asuitable controlled unitary operation (here quantum Tof-foli gate), applied the operation which should be repli-cated to the auxiliary qubit, and then disentangled theauxiliary qubit from the signal qubits by a suitable quan-tum measurement. Intriguingly, this procedure convertsthe replicated single-qubit phase gate to a two-qubitcontrolled-phase gate so it adds a control to an arbitraryunknown phase gate U ( φ ).Our work paves the way towards experimental imple-mentations of even more advanced schemes for quantumgate replication. The demonstrated approach based onthe Toffoli gate can be extended to arbitrary number ofsignal qubits M and copies of unitary phase operation N . For any M and N , the transformation (3) could beimplemented by a sequence of several ( M + N )-qubit gen-eralized Toffoli gates, where each generalized Tofolli gatewould induce nontrivial bit flips on the subspace of aux-iliary qubits for one particular basis state of the signalqubits and would behave as an identity operation for allother basis states of the signal qubits. However, for therelevant scenario N ≥ √ M the number of gates wouldgrow exponentially with the number of qubits. Therefore,it is likely that the constituent generalized quantum Tof-foli gates would have to exhibit fidelities exponentiallyclose to 1 to ensure the desired high-fidelity performanceof the superreplication protocol. Acknowledgments
This work was supported by the Czech Science Foun-dation (GA13-20319S).
Appendix: Plots of all reconstructed quantumprocess matrices
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