Experimental Determination of Ramsey Numbers

(Left to Right) Zhengbing Bian, Fabian Chudak, William G. Macready, Lane Clark

Authors: Zhengbing Bian¹, Fabian Chudak¹, William G. Macready¹, 
Lane Clark², Frank Gaitan³

Affiliation:
¹D-Wave Systems Inc., Burnaby, British Columbia, Canada
²Dept. of Mathematics, Southern Illinois University, Carbondale, Illinois, USA
³Laboratory for Physical Sciences, College Park, Maryland, USA

                                                                       Frank Gaitan 

Ramsey numbers are most easily introduced through the Party Problem. Here a host is assembling a guest list of N people. She wonders whether the list might contain a group of m guests who are mutual friends, or a group of n guests who are mutual strangers. Surprisingly, it can be proved [1] that there exists a threshold size R(m,n) for the guest list such that: (i) if N ≥ R(m,n), then all guest lists are sure to contain a clique of m mutual friends or one of n mutual strangers, and (ii) if N < R(m,n), then a guest list exists which contains no group of m mutual friends or n mutual strangers. The threshold value R(m,n) is known as a two-color Ramsey number.

It proves fruitful to convert the N-person guest list into an N-vertex graph by associating each guest with a distinct vertex in the graph. We draw a red (blue) edge between two vertices if the two corresponding people know (do not know) each other. Thus, if m (n) guests all know (do not know) each other, there will be a red (blue) edge connecting any two of the m (n) corresponding vertices. The m (n) mutual friends (strangers) are said to form a red m-clique (blue n-clique). In this way, each N-person guest list is associated with a particular red/blue coloring of the edges of a complete N-vertex graph (in which every pair of vertices is joined by an edge). The threshold result for the Party Problem is now the following statement about graphs: if N ≥ R(m,n), then every red-blue edge coloring of a complete N-vertex graph will contain either a red m-clique or a blue n-clique, while if N < R(m,n), a red/blue coloring exists which has no red m-clique or blue n-clique. The use of two colors is why R(m,n) is called a two-color Ramsey number. Figure 1 shows a 6-vertex graph that contains two blue 3-cliques and no red 3-cliques.

Figure 1: A 6-vertex graph representing a Guest List in which there are: (i) two cliques of three people that are mutual strangers: Guests 1, 2, and 3, and Guests 4, 5, and 6; and (ii) no cliques of three mutual friends.

As m and n increase, the threshold R(m,n) grows extremely rapidly, and even for quite small values of m and n, becomes extraordinarily difficult to compute. In fact, for m, n ≥ 3, only nine values of R(m,n) are currently known [2]. To dramatize the difficulty of calculating R(m,n), Spencer [3] recounts a famous vignette due to Paul Erdos in which aliens have invaded earth and threaten to destroy it in one year unless humans can correctly determine R(5,5). Erdos noted that this could probably be done if all the best minds and most powerful computers were focused on the task. In this case, the appropriate response to the alien demand would be to get busy. However, Erdos pointed out that if the aliens had instead demanded R(6,6), earthlings should immediately launch a first-strike on the aliens as there is little hope of meeting their demand.

In 2012 a quantum algorithm for computing two-color Ramsey numbers appeared which was based on adiabatic quantum evolution [4]. The essential step in constructing the Ramsey number algorithm was to transform a Ramsey number computation into an optimization problem whose solution is an N-vertex graph that has the smallest total count of red m-cliques and blue n-cliques. The optimization cost function assigns to each graph the total count of red m-cliques and blue n-cliques that it contains. Thus, if N < R(m,n), an optimal graph will have zero cost, corresponding to a guest list with no m-clique of friends or n-clique of strangers. On the other hand, if N ≥ R(m,n), then an optimal graph has non-zero cost since all guest lists will have a red m-clique or blue n-clique. The Ramsey number quantum algorithm associates each graph with a unique quantum state of the qubit-register and constructs the qubit interactions so that the energy of this state (the Ramsey energy) is the value of the graph’s cost function. By design, the minimum Ramsey energy is the minimum value of the cost function, and the associated quantum state identifies a minimum cost graph which is thus a solution of the optimization problem.

The adiabatic quantum evolution algorithm [5] is used to transform a fiducial state of the qubit register into a state with lowest Ramsey energy. The Ramsey number algorithm implements the following iterative procedure. It starts with graphs with N < R(m,n), and carries out the above described adiabatic quantum evolution. At the end of the evolution, the qubit state is measured and the Ramsey energy is determined from the measurement result. In the adiabatic limit, for N < R(m,n), the measured Ramsey energy will be zero. The number of vertices N is then increased by one, the adiabatic evolution and measurement is re-run, and the Ramsey energy re-computed. This process is repeated until a non-zero Ramsey energy is first found. In the adiabatic limit, the final value of N is the Ramsey number R(m,n) since it is the smallest N value for which all graphs have a non-zero cost and thus contain a red m-clique or blue n-clique. Ref. [5] tested the algorithm via numerical simulations and it correctly found the Ramsey numbers R(3,3) = 6 and R(m,2) = m for 5 ≤ m ≤ 7.

Figure 2: Image of the harness used to lower the D-Wave One 128 qubit chip into a dilution refrigerator. The chip can be seen at the bottom of the harness.

Recently, the Ramsey number quantum algorithm was successfully implemented on a D-Wave One quantum annealing device [6]. Quantum annealing [7] can be used to drive the adiabatic dynamics of the Ramsey number algorithm even at finite temperature and in the presence of decoherence. The D-Wave One processor is a chip that contains 128 rf SQUID flux qubits. The chip is shown in Figure 2 at the end of a harness which is used to place it in a dilution refrigerator that operates at 20mK.

The chip used in the Ramsey number experiments contained 106 useable qubits. Figure 3 shows the chip layout, where the grey circles correspond to useable qubits, and the white circles to qubits that could not be calibrated into the desired tolerance range and so were not used. The edges joining qubits represent couplers that allow the qubits to interact and which can take on programmed values.

Figure 3: Layout of the 128 qubit chip. The chip architecture is a 4x4 array of unit cells, with each unit cell containing 8 qubits. Qubits are labeled from 1 to 128, and edges between qubits indicate couplers which may take programmable values. Grey qubits indicate usable qubits, while white qubits indicate qubits which, due to fabrication defects, could not be calibrated to operating tolerances and were not used. All Ramsey number experiments were done on a chip with 106 usable qubits.

Figure 4 shows an 8-qubit unit cell in the chip. Each red loop highlights one of the rf SQUID flux qubits. Programming the Ramsey number cost function onto the D-Wave One chip requires embedding the optimization variables into the qubit architecture in a way that respects the qubit-qubit couplings on the chip. Figure 5 shows the embedding used to determine the Ramsey number R(8,2). Once the cost function for a given Ramsey number has been programmed on to the chip, 100,000 quantum annealing sweeps and measurements were carried out, and the resulting Ramsey energies binned into a histogram.

In all cases considered, the lowest Ramsey energy found was identical to the minimum Ramsey energy which was known from previous work [4]. The protocol for the Ramsey number algorithm was carried out and the value of N at which the minimum Ramsey energy first became non-zero was identified with the Ramsey number R(m,n). The experiments correctly determined that R(3,3) = 6, and R(m,2) = m for 4 ≤ m ≤ 8. Although all Ramsey numbers found in Ref. [6] correspond to known Ramsey numbers, it is possible that future hardware generations may be large enough to allow unknown Ramsey numbers to be found.

Figure 4: A single 8 qubit unit cell. Each red loop highlights an rf SQUID flux qubit.

Figure 5: Embedding used to compute the Ramsey number R(8,2) that produced the needed qubit couplings. In low energy states, like-colored qubits have the same Bloch vector and constitute a single effective qubit. This allows an indirect coupling of qubits that are not directly coupled on the chip.

References:
[1] Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer, "Ramsey Theory" (Wiley, NY, 1990).
[2] See Wikipedia page on Ramsey's_theorem for a list of all known two-color Ramsey numbers, as well as bounds on R(m,n) when its value is not known, for m, n ≤ 10.
[3] Joel Spencer, "Ten Lectures on the Probabilistic Method", 2nd ed. (SIAM, Philadelphia, 1994).
[4] Frank Gaitan and Lane Clark, "Ramsey Numbers and Adiabatic Quantum Computing", Physical Review Letters 108, 010501 (2012). Abstract.
[5] Edward Farhi, Jeffrey Goldstone, Sam Gutmann, Michael Sipser, "Quantum Computation by Adiabatic Evolution", arXiv:quant-ph/0001106 (2000).
[6] Zhengbing Bian, Fabian Chudak, William G. Macready, Lane Clark, and Frank Gaitan, "Experimental Determination of Ramsey Numbers", Physical Review Letters, 111, 130505 (2013). Abstract.
[7] Tadashi Kadowaki and Hidetoshi Nishimori, "Quantum annealing in the transverse Ising model", Physical Review E 58, 5355 (1998). Abstract; Giuseppe E. Santoro, Roman Martoňák, Erio Tosatti, Roberto Car, "Theory of Quantum Annealing of an Ising Spin Glass", Science 295, 2427 (2002). Abstract.

Entanglement-enhanced Atom Interferometer with High Spatial Resolution

(From left to right) Philipp Treutlein, Roman Schmied, and Caspar Ockeloen

Authors: Caspar Ockeloen, Roman Schmied, Max F. Riedel, Philipp Treutlein

Affiliation: Department of Physics, University of Basel, Switzerland

Link to Quantum Atom Optics Lab, Treutlein Group >>

Interferometry is the cornerstone of most modern precision measurements. Atom interferometers, making use of the wave-like nature of matter, allow for ultraprecise measurements of gravitation, inertial forces, fundamental constants, electromagnetic fields, and time [1,2]. A well-known application of atom interferometry is found in atomic clocks, which provide the definition of the second. Most current atom interferometers operate with a large number of particles, which provides high precision but limited spatial resolution. Using a small atomic cloud as a scanning probe interferometer would enable new applications in electromagnetic field sensing, surface science, and the search for fundamental short-range interactions [2].

Past 2Physics article by Philipp Treutlein:
May 09, 2010: "Interface Between Two Worlds -- Ultracold atoms coupled to a micromechanical oscillator"
by Philipp Treutlein, David Hunger, Stephan Camerer.

In an atom interferometer, the external (motional) or internal (spin) state of atoms is coherently split and allowed to follow two different pathways. During the interrogation time T, a phase difference between the paths is accumulated, which depends on the quantity to be measured. When the paths are recombined, the wave-character of the atoms gives rise to an interference pattern, from which the phase can be determined. To measure this interference, the number of atoms in each output state is counted. Here the particle-character of the atoms is revealed, as the measurement process randomly projects the wave function of each atom into a definite state. When operating with an ensemble of N uncorrelated (non-entangled) atoms, the binomial counting statistics limits the phase uncertainty of the interferometer to 1/√N, the standard quantum limit (SQL) of interferometric measurement.

It is possible to overcome the SQL by making use of entanglement between the atoms [3]. Using such quantum correlations, the measurement outcome of each atom can depend on that of the other atoms. If used in a clever way, the phase uncertainty of an interferometer can be reduced below the SQL, in theory down to the ultimate Heisenberg limit of 1/N. Such entanglement-enhanced interferometry is in particular useful in situations where the number of atoms is limited by a physical process and the sensitivity can no longer be improved by simply increasing N. One such scenario is when high spatial resolution is desired. The number of atoms in a small probe volume is fundamentally limited by density-dependent losses due to collisions. As more atoms are added to this volume, the collision rate increases, and eventually any additional atoms are simply lost from the trap before the interferometer sequence has completed. This sets a tight limit on both the phase uncertainty and the maximum interrogation time T.

Fig. 1. Experimental setup. a) Central region of the atom chip showing the atomic probe (blue, size to scale) and the scanning trajectory we use. The probe is used to measure the magnetic near-field potential generated by an on-chip microwave guide (microwave currents indicated by arrows). A simulation of the potential is shown in red/yellow. b) Photograph of the atom chip, mounted on its ultra-high vacuum chamber.

In a recent paper [4] we have demonstrated a scanning-probe atom interferometer that overcomes the SQL using entanglement. Our interferometer probe is a Bose-Einstein condensate (BEC) on an atom chip, a micro-fabricated device with current-carrying wires that allow magnetic trapping and accurate positioning of neutral atoms close to the chip surface [5]. A schematic view of the experiment is shown in figure 1. We use N=1400 Rubidium-87 atoms, trapped in a cloud of 1.1 x 1.1 x 4.0 micrometers radius, 16 to 40 micrometers from the surface. Two internal states of the atoms are used as interferometric pathways, and the pathways are split and recombined using two-photon microwave and radio frequency pulses. At the end of the interferometer sequence, we count the atoms in each output state with sensitive absorption imaging, with a precision of about 5 atoms.

We create entanglement between the atoms by making use of collisions naturally present in our system. When two atoms collide, both atoms obtain a phase shift depending on the state of the other atom, thus creating quantum correlations between the two. Normally, the effect of these collisions is negligible in our experiment, as the phase shift due to collisions between atoms in the same state are almost completely canceled out by collisions where each atom is in a different state. We can turn on the effect of collisions by spatially separating the two states, such that collisions between states do not occur. When, after some time, we recombine the two states, collisional phase shifts are effectively turned off during the subsequent interrogation time of the interferometer.

The performance of our interferometer is shown in figure 2, measured at 40 micrometer from the chip surface. It has a sensitivity of 4 dB in variance below to SQL, and improved sensitivity is maintained for up to T = 10 ms of interrogation time, longer than in previous experiments [6,7,8]. We demonstrate the scanning probe interferometer by transporting the entangled atoms between 40 and 16 micrometer from the atom chip surface, and measuring a microwave near field potential at each location. The microwave potential is created by wires on our atom chip, and is also used for generation of the entangled state. As shown in figure 3, our scanning probe interferometer operates on average 2.2 dB below the SQL, demonstrating that the entanglement partially survives being transported close to the chip surface, which takes 20 ms of transport time.

Fig. 2. Interferometer performance operating at a single position for different interrogation times. Plotted is the variance relative to the standard quantum limit (SQL). The entanglement-enhanced interferometer (blue diamonds) operates about 4 dB below the SQL, whereas the non-entangled interferometer (red, coherent state) operates close to the SQL. For T > 10 ms, both experiments are limited by technical noise. The inset shows a typical interference fringe, with a fringe contrast of (98.1 ± 0.2)%.

The scanning probe measurement presented here corresponds to a microwave magnetic field sensitivity of 2.4 µT in a single shot of the experiment (cycle time ~ 11 s). The sensitivity shown in figure 2 corresponds to 23 pT for an interrogation time of 10 ms. This sensitivity is obtained with a probe volume of only 20 µm³. Our interferometer bridges the gap between vapor cell magnetometers, which achieve subfemtotesla sensitivity at the millimeter to centimeter scale [9,10] but do not have the spatial resolution needed to resolve near-field structures on microfabricated devices, and nitrogen vacancy centers in diamond, which are excellent magnetometers at the nanometer scale but currently offer lower precision in the micrometer regime [11].

Fig. 3. Scanning probe interferometer. a) Phase shift due to the microwave near-field potential measured at different positions. The dashed line is a simulation of the potential. b) Interferometer performance for the same measurement. At all positions, the interferometer operates below the SQL. These measurements were done with an interrogation time of T = 100 µs, during which the microwave near-field was pulsed on for 80 µs.

In conclusion, we have experimentally demonstrated a scanning-probe atom interferometer operating beyond the standard quantum limit, and used it for the measurement of a microwave near-field. High-resolution measurements of microwave near-fields are relevant for the design of new microwave circuits for use in communication technology [12]. This is the first demonstration of entanglement-enhanced atom interferometry with a high spatial resolution scanning probe, and promises further high-resolution sensing and measurement applications.

References:
[1] Alexander D. Cronin, Jörg Schmiedmayer, David E. Pritchard, "Optics and interferometry with atoms and molecules", Review of Modern Physics, 81, 1051 (2009). Abstract.
[2] J. Kitching, S. Knappe, and E.A. Donley, "Atomic Sensors – A Review", IEEE Sensors Journal, 11, 1749 (2011). Abstract.
[3] Vittorio Giovannetti, Seth Lloyd, Lorenzo Maccone, "Advances in quantum metrology", Nature Photonics, 5, 222 (2011). Abstract.
[4] Caspar F. Ockeloen, Roman Schmied, Max F. Riedel, Philipp Treutlein, "Quantum Metrology with a Scanning Probe Atom Interferometer", Physical Review Letters, 111, 143001 (2013). Abstract.
[5] Max F. Riedel, Pascal Böhi, Yun Li, Theodor W. Hänsch, Alice Sinatra, Philipp Treutlein, "Atom-chip-based generation of entanglement for quantum metrology", Nature, 464, 1170 (2010). Abstract.
[6] C. Gross, T. Zibold, E. Nicklas, J. Estève, and M.K. Oberthaler, "Nonlinear atom interferometer surpasses classical precision limit", Nature, 464, 1165 (2010). Abstract.
[7] Anne Louchet-Chauvet, Jürgen Appel, Jelmer J Renema, Daniel Oblak, Niels Kjaergaard, Eugene S Polzik, "Entanglement-assisted atomic clock beyond the projection noise limit", New Journal of Physics, 12, 065032 (2010). Abstract.
[8] Ian D. Leroux, Monika H. Schleier-Smith, and Vladan Vuletić, "Orientation-Dependent Entanglement Lifetime in a Squeezed Atomic Clock", Physical Review Letters, 104, 250801 (2010). Abstract.
[9] R. Mhaskar, S. Knappe, and J. Kitching, "A low-power, high-sensitivity micromachined optical magnetometer", Applied Physics Letters, 101, 241105 (2012). Abstract.
[10] W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, E. S. Polzik, "Quantum Noise Limited and Entanglement-Assisted Magnetometry", Physical Review Letters, 104, 133601 (2010). Abstract.
[11] S. Steinert, F. Dolde, P. Neumann, A. Aird, B. Naydenov, G. Balasubramanian, F. Jelezko, J. Wrachtrup, "High sensitivity magnetic imaging using an array of spins in diamond", Review of Scientific Instruments, 81, 043705 (2010). Abstract.
[12] S. Sayil, D.V. Kerns, jr. and S.E. Kerns, "Comparison of contactless measurement and testing techniques to a all-silicon optical test and characterization method", IEEE Trans. Instrum. Meas. 54, 2082 (2005). Abstract. 

Deterministic Quantum Teleportation with Feed-Forward in a Solid State System

[From left to right] Andreas Wallraff, Christopher Eichler, Yves Salathe, Markus Oppliger, Philipp Kurpiers, Lars Steffen

Authors: Lars Steffen, Yves Salathe, Markus Oppliger, Philipp Kurpiers, Matthias Baur^*, Christian Lang, Christopher Eichler, Gabriel Puebla-Hellmann, Arkady Fedorov^*, Andreas Wallraff

Affiliation: Department of Physics, ETH Zurich, Switzerland

^*Present address: ARC Centre for Engineered Quantum Systems, University of Queensland, Brisbane, Australia

Link to QUDEV-Lab >>

Transferring the state of an information carrier between two parties is an essential primitive in both classical and quantum communication and information processing. Quantum teleportation [1] describes the concept of transferring an unknown quantum state from a sender to a physically separated receiver without transmitting the physical carrier of information itself. To do so, teleportation makes use of the non-local correlations provided by an entangled pair shared between the sender and the receiver and the exchange of classical information.

In our recent publication [2] we report the first teleportation of information in a solid state system. We use a chip-based superconducting circuit architecture [3, 4] (Fig. 1) with three superconducting transmon qubits [5] and three superconducting coplanar waveguide resonators [6] which serve as a quantum bus [7, 8] and allow to perform qubit state readout [9, 10]. We have realized the full deterministic quantum teleportation protocol using quantum-limited parametric amplifiers [11] for single-shot readout, a crossed quantum bus technology and flexible real-time digital electronics.

Figure 1: In our chip-based superconducting circuit three qubits are coupled to three resonators to realize deterministic quantum teleportation.

The success of the teleportation protocol in every instance with unit fidelity is counterintuitive from a classical point of view [12]. The receiver’s quantum bit (qubit) does not interact with any other qubit after it has been entangled with one of two qubits in the sender’s possession. The input state (|ψ_in>) is prepared at a later time at the sender. The classical information sent by the sender is not sufficient to recreate |ψ_in> perfectly at the receiver. Indeed, assuming no entanglement between sender and receiver one can replicate the sender’s state at best with a process fidelity of 1/2. To always recover the original state |ψ_in> the sender performs a measurement in the basis of the Bell states, which projects the two qubits in the sender’s possession randomly onto one of the four Bell states. As a consequence the receiver’s qubit is projected instantaneously into a state related to |ψ_in> without ever having interacted with the sender’s qubit. The receiver’s qubit only differs from the input state by a single-qubit rotation which depends on the four possible measurement results. In the final step, the sender communicates the Bell measurement result as two bits of classical information via a classical channel and therefore the receiver can always obtain the original input state |ψ_in>. This final step is frequently referred to as feed-forward.

Figure 2: a) Standard protocol of quantum teleportation. The protocol starts with the preparation of a Bell state between Q2 and Q3 (blue box) followed by the preparation of an arbitrary state |ψ_in> (green box) and a Bell state measurement of Q1 and Q2 (red box). The classical information extracted by the measurement of Q1 and Q2 is transferred to the receiver to perform local gates conditioned on the measurement outcomes. At the end of the protocol Q3 is in a state |ψ_out> which ideally is identical to |ψ_in> (also colored in green). Here, H is the Hadamard gate, X and Z are Pauli matrices σ_x and σ_z, respectively. The cnot-gate is represented by a vertical line between the control qubit (•) and the target qubit (⊕). b) The protocol implemented in the experiment presented here uses controlled-phase gates indicated by vertical lines between the relevant qubits (•), and single qubit rotations R^θ_±y of angle θ about the ±y-axis. To finalize the teleportation we either post-select on any single one of the four measurement outcomes (00, 01, 10 and 11) acquired in a single shot, or we deterministically use all four outcomes, which we then may use to implement feed-forward. The feed-forward operators R^π_x and R^π_y are applied to Q3 conditioned on the four measurement outcomes according to the table presented in the framed box.

Our implementation of the protocol (Fig. 2b) uses single qubit rotations and controlled-phase gates [13, 14] and is equivalent to the original protocol shown in Fig. 2a. The teleportation process succeeds with order unit probability for any input state, as we reliably prepare entangled states as a resource and are able to distinguish all four maximally entangled Bell states in a single measurement.

Figure 3: The solid bars show the experimentally obtained process matrix describing the state transfer from the sender to the receiver. The ideal process matrix, which is the identity operation, is shown in wire frames.

To identify simultaneously the four outcomes of the Bell-state measurement in a deterministic way we use two-qubit single-shot read-out [15, 16] for which we achieve a success probability of (81.8±0.5) %. In the teleportation protocol we analyze the Bell-state measurement in real time by using fast electronics based on a field programmable gate array (FPGA) and realize the feed-forward step in about 500 ns. We have achieved an average process fidelity of (62.2±0.3) % for the full quantum teleportation algorithm, which is clearly above the classical threshold (Fig. 3). We have demonstrated teleportation at a rate of 10,000 quantum bits per second between two macroscopic systems separated by 6 mm.

The low transmission loss of superconducting waveguides is likely to enable the range of this and other schemes to be extended to significantly larger distances, enabling tests of non-locality and the realization of elements for quantum communication at microwave frequencies. The demonstrated feed-forward may also find application in error correction schemes.

References:
[1] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters. "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels". Physical Review Letters, 70, 1895 (1993). Abstract.
[2] L. Steffen, Y. Salathe, M. Oppliger, P. Kurpiers, M. Baur, C. Lang, C. Eichler, G. Puebla-Hellmann, A. Fedorov, and A. Wallraff. "Deterministic quantum teleportation with feed-forward in a solid state system". Nature, 500, 319 (2013). Abstract.
[3] A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf. "Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics". Nature, 431, 162 (2004). Abstract.
[4] F. Helmer, M. Mariantoni, A. G. Fowler, J. von Delft, E. Solano, and F. Marquardt. "Cavity grid for scalable quantum computation with superconducting circuits". Europhysics Letters, 85, 50007 (2009). Abstract.
[5] Jens Koch, Terri M. Yu, Jay Gambetta, A. A. Houck, D. I. Schuster, J. Majer, Alexandre Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf. "Charge-insensitive qubit design derived from the Cooper pair box". Physical Review A, 76, 042319 (2007). Abstract.
[6] M. Göppl, A. Fragner, M. Baur, R. Bianchetti, S. Filipp, J. M. Fink, P. J. Leek, G. Puebla, L. Steffen, and A. Wallraff. "Coplanar waveguide resonators for circuit quantum electrodynamics". Journal of Applied Physics, 104, 113904 (2008). Abstract.
[7] J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf. "Coupling superconducting qubits via a cavity bus". Nature, 449, 443 (2007). Abstract.
[8] Mika A. Sillanpää, Jae I. Park, Raymond W. Simmonds. "Coherent quantum state storage and transfer between two phase qubits via a resonant cavity". Nature, 449, 438 (2007). Abstract.
[9] R. Bianchetti, S. Filipp, M. Baur, J. M. Fink, M. Göppl, P. J. Leek, L. Steffen, A. Blais, and A. Wallraff. "Dynamics of dispersive single-qubit readout in circuit quantum electrodynamics". Physical Review A, 80, 043840 (2009). Abstract.
[10] S. Filipp, P. Maurer, P. J. Leek, M. Baur, R. Bianchetti, J. M. Fink, M. Göppl, L. Steffen, J. M. Gambetta, A. Blais, and A. Wallraff. "Two-qubit state tomography using a joint dispersive readout". Physical Review Letters, 102, 200402 (2009). Abstract,
[11] Bernard Yurke and Eyal Buks. "Performance of cavity-parametric amplifiers, employing Kerr nonlinearites, in the presence of two-photon loss". Journal of Lightwave Technology, 24, 5054 (2006). Abstract.
[12] S. Massar and S. Popescu. "Optimal extraction of information from finite quantum ensembles". Physical Review Letters, 74, 1259 (1995). Abstract.
[13] Frederick W. Strauch, Philip R. Johnson, Alex J. Dragt, C. J. Lobb, J. R. Anderson, and F. C. Wellstood. "Quantum logic gates for coupled superconducting phase qubits". Physical Review Letter, 91, 167005 (2003). Abstract.
[14] L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf. "Demonstration of two-qubit algorithms with a superconducting quantum processor". Nature, 460, 240 (2009). Abstract.
[15] François Mallet, Florian R. Ong, Agustin Palacios-Laloy, François Nguyen, Patrice Bertet, Denis Vion & Daniel Esteve. "Single-shot qubit readout in circuit quantum electrodynamics". Nature Physics, 5, 79 (2009). Abstract.
[16] R. Vijay, D. H. Slichter, and I. Siddiqi. "Observation of quantum jumps in a superconducting artificial atom". Physical Review Letters, 106, 110502 (2011). Abstract.

Hybrid Quantum Teleportation

[From Left to Right] Shuntaro Takeda, Maria Fuwa, and Akira Furusawa

Authors: Shuntaro Takeda, Maria Fuwa, and Akira Furusawa

Affiliation: Department of Applied Physics, School of Engineering, University of Tokyo, Japan

Link to Furusawa Laboratory >>

The principles of quantum mechanics allow us to realize ultra-high-capacity optical communication and ultra-high-speed quantum computation beyond the limits of current technology. One of the most fundamental steps towards this goal is to transfer quantum bits (qubits) carried by photons through “quantum teleportation” [1]. Quantum teleportation is the act of transferring qubits from a sender to a spatially distant receiver by utilizing shared entanglement and classical communications.

After its original proposal in 1993 [1], a research group in Austria succeeded in its realization in 1997 [2]. However, this scheme involved several deficiencies. One is its low transfer efficiency, estimated to be far below 1%. This is due to the probabilistic nature of entanglement generation and the joint measurement of two photons. This scheme also required post-selection of the successful events by measuring the output qubits after teleportation [3]. The transferred qubits are destroyed in this process, and thus cannot be used for further information processing. Various other related experiments have been reported thus far, but most withhold the same disadvantages. This problem has been a major limitation in the development of optical quantum information processing.

In our recent publication [4], we demonstrated “deterministic” quantum teleportation of photonic qubits for the first time. That is, photonic qubits are always teleported in each attempt, in contrary to the former probabilistic scheme. In addition, it does not require post-selection of the successful events. The success of the experiment lies in a hybrid technique of photonic qubits and continuous-variable quantum teleportation [5,6,7]; this required the combination of two conceptually different and previously incompatible approaches.

Figure 1: Concept of our hybrid technique for quantum teleportation. Single-photon-based qubits are combined with continuous-variable quantum teleportation to transfer optical waves.

Continuous-variable quantum teleportation, first demonstrated in 1998 [7], has long been used to teleport the amplitude and phase signals of optical waves, rather than photonic qubits. The advantage of continuous-variable teleportation is its deterministic success due to the on-demand availability of entangled waves and the complete joint measurement of two waves. However, its application to photonic qubits had long been hindered by experimental incompatibilities: typical pulsed-laser-based qubits have a broad frequency bandwidth that is incompatible with the original continuous-wave-based continuous-variable teleporter, which works only on narrow frequency sidebands. We overcame this incompatibility by developing an innovative technology: a broadband continuous-variable teleporter [8] and a narrowband qubit compatible with that teleporter [9].

Figure 2: Configuration of the teleportation experiment. Laser sources and non-linear optical processes supply the qubit and the required entanglement. More than 500 mirrors and beam splitters constitute the teleportation circuit.

This hybrid technique enabled the realization of completely deterministic and unconditional quantum teleportation of photonic qubits. The transfer accuracy (fidelity) ranged from 79 to 82 percent in four different qubits. Another strength of our hybrid scheme lies in the fact that the qubits were teleported much more efficiently than the previous scheme, even with low degrees of entanglement. This is a decisive breakthrough in the field of optical teleportation 16 years after the first experimental realizations.

References:
[1] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters, “Teleporting an Unknown Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels”, Physical Review Letters, 70, 1895 (1993). Abstract.
[2] Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter, Anton Zeilinger, “Experimental quantum teleportation”, Nature, 390, 575 (1997). Abstract.
[3] Samuel L. Braunstein and H. J. Kimble, “A posteriori teleportation”, Nature 394, 840 (1998). Abstract.
[4] Shuntaro Takeda, Takahiro Mizuta, Maria Fuwa, Peter van Loock, Akira Furusawa, “Deterministic quantum teleportation of photonic quantum bits by a hybrid technique”, Nature 500, 315 (2013). Abstract.
[5] Lev Vaidman, “Teleportation of quantum states”, Physical Review A 49, 1473 (1994). Abstract.
[6] Samuel L. Braunstein and H. J. Kimble, "Teleportation of Continuous Quantum Variables", Physical Review Letters, 80, 869 (1998). Abstract.
[7] A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional Quantum Teleportation”, Science 282, 706 (1998). Abstract.
[8] Noriyuki Lee, Hugo Benichi, Yuishi Takeno, Shuntaro Takeda, James Webb, Elanor Huntington, Akira Furusawa, “Teleportation of Non-Classical Wave-Packets of light”, Science 332, 330 (2011). Abstract.
[9] Shuntaro Takeda, Takahiro Mizuta, Maria Fuwa, Jun-ichi Yoshikawa, Hidehiro Yonezawa, Akira Furusawa, “Generation and eight-port homodyne characterization of time-bin qubits for continuous-variable quantum information processing”, Physical Review A 87, 043803 (2013). Abstract.

Coherently Manipulating Nanomechanical Oscillators

[From Left to Right] Hajime Okamoto, Imran Mahboob, Hiroshi Yamaguchi

Authors: Hajime Okamoto, Imran Mahboob, and Hiroshi Yamaguchi

Affiliation: NTT Basic Research Laboratories, Atsugi-shi, Kanagawa, Japan

Semiconductor nanomechanical oscillators enable the pursuit of new physical phenomenon that can only be observed through the tiny mechanical displacement as well as enabling the development of nanoscience and nanotechnology. These systems with sharp mechanical resonances can be manipulated by electrical or optical means permitting a wide range of applications such as metrology and information processing [1].

Coupling two such nanomechanical oscillators has recently emerged as a subject of interest. This is because the sympathetic oscillation dynamics in the coupled system expand the potential applications of nanomechanical objects such as highly precise sensors, high-Q band-pass filters, signal amplifier, and even logic gates [2-5]. However, an obstacle to the further development of this architecture arises from the usually weak elastic (or electric) coupling between the nanomechanical components. This limits the ability to coherently transfer the vibration energy between the oscillators within the oscillator’s ring-down time. In other words, the coherent manipulation of the vibrational states, e.g., Rabi cycle and Ramsey fringe as in two-level quantum systems [6], is highly challenging in this coupled phonon system.

In our recent publication [7], we demonstrated coherent coupling of nanomechanical oscillators. This was realized in two geometrically interconnected GaAs doubly-clamped beams [Fig. 1(a)], where the frequency of beam R is higher than that of beam L as shown in Fig. 1(b). Because of the frequency mismatch and the weak mechanical coupling, coherent energy exchange between the two beams is not possible by the geometric interconnection alone.

Figure 1. (a) Schematic drawing of the sample and the piezoelectric effect. The piezoelectric effect enables harmonic driving, pumping, and detection of the mechanical motion via gate voltage. (b) Schematic drawing of the mechanical resonance for beam L and beam R. The frequency of beam R is higher than that of beam L (293.93 kHz) by 440 Hz. The coherent energy exchange between the two beams is achieved by applying the pumping voltage to beam L with the frequency difference between the two beams. (c) Schematic of the pumping protocol in a mass and spring model.

Coherent energy exchange can be achieved by dynamically coupling the mechanical oscillations of the two beams. This is realized by periodically modulating the spring constant of one beam at the frequency difference between the two beams [Fig. 1(c)]. This periodic modulation, namely pumping, can be induced by applying gate voltage via the piezoelectric effect in this sample [Fig. 1(a)]. This pumping enables strong vibrational coupling, leading to the cyclic (Rabi) oscillations between the two vibrational states (the beam-L state and the beam-R state) on the Bloch sphere (Fig. 2). The Rabi cycle period, i.e., the coupling strength, is fully adjustable by changing the pump amplitude via the gate voltage. As a result, the vibration energy can be quickly transferred from one beam to the other enabling the vibration of the original beam to be switched off on a time-scale orders of magnitude shorter than its ring-down time. This quick energy transfer to the adjacent oscillator opens up the prospect of high-speed repetitive operations for sensors and logics using nanomechanical systems [8,9].

Figure 2. Schematic drawing showing the evolution of the coupled mechanical oscillators on the Bloch sphere when undergoing a Rabi cycle.

In terms of phonon populations, the present system with sub-megahertz frequency is still in the classical regime with large mode occupation. However, the above technique could also be extended to gigahertz frequency mechanical oscillators in which the average phonon number in the mechanical modes falls below one at cryogenic temperatures [10]. This in turn leads to the exciting possibility of phononic quantum bits and entanglement between distinct macroscopic mechanical elements [11].

References:
[1] A. N. Cleland, “Foundations of Nanomechanics” (Springer-Verlag Berlin Heidelberg New York, 2003). [2] Matthew Spletzer, Arvind Raman, Alexander Q. Wu, Xianfan Xu, Ron Reifenberger, “Ultrasensitive mass sensing using mode localization in coupled microcantilevers”, Applied Physics Letters, 88, 254102 (2006). Abstract.
[3] F. D. Bannon, J. R. Clark, and C. T.-C. Nguyen, “High-Q HF microelectromechanical filters”, IEEE J. Solid-State Circuits 35, 512-526 (2000). Abstract.
[4] R. B. Karabalin, Ron Lifshitz, M. C. Cross, M. H. Matheny, S. C. Masmanidis, and M. L. Roukes, “Signal amplification by sensitive control of bifurcation topology”, Physical Review Letters, 106, 094102 (2011). Abstract.
[5] Sotiris C. Masmanidis, Rassul B. Karabalin, Iwijn De Vlaminck, Gustaaf Borghs, Mark R. Freeman, Michael L. Roukes, “Multifunctional nanomechanical systems via tunably coupled piezoelectric actuation”, Science, 317, 780-783 (2007). Abstract.
[6] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson, A. C. Gossard, “Coherent manipulation of coupled electron spins in semiconductor quantum dots”, Science, 309, 2180-2184 (2005). Abstract.
[7] Hajime Okamoto, Adrien Gourgout, Chia-Yuan Chang, Koji Onomitsu, Imran Mahboob, Edward Yi Chang, Hiroshi Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators”, Nature Physics, 9, 480-484 (2013). Abstract.
[8] Hiroshi Yamaguchi, Hajime Okamoto, and Imran Mahboob, “Coherent control of micro/nanomechanical oscillation using parametric mode mixing”, Applied Physics Express, 5, 014001 (2012). Abstract.
[9] I. Mahboob, E. Flurin, K. Nishiguchi, A. Fujiwara, H. Yamaguchi, “Interconnect-free parallel logic circuits in a single mechanical resonator”, Nature Communications, 2, 198 (2011). Abstract.
[10] A. D. O’Connell, M. Hofheinz, M. Ansmann, Radoslaw C. Bialczak, M. Lenander, Erik Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, John M. Martinis, A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator”, Nature, 464, 697-703 (2010). Abstract.
[11] Simon Rips and Michael J. Hartmann, “Quantum information processing with nanomechanical qubits”, Physical Review Letters, 110, 120503 (2013). Abstract.

Studying Light Pulses By Counting Photons

Elizabeth A. Goldschmidt (left) 
and Alan Migdall (right) 
[Photo courtesy: JQI/NIST/University of Maryland]

The photodetectors in Alan Migdall’s lab often see no light at all, and that’s a good thing since he and his colleagues at the Joint Quantum Institute (JQI) perform physics experiments that require very little light, the better to study subtle quantum effects. The bursts of light they observe typically consist of only one or two photons--- the particle form of light---or (statistically speaking) even less than one photon. Their latest achievement is to develop a new way of counting photons to understand the sources and modes of light in modern physics experiments.

Migdall’s lab, located at the National Institute for Standards and Technology (NIST), is just outside Washington, DC in the US. The new light-measuring protocol is summarized in a recent issue of the journal Physical Review A [1]. The work reported there was performed in collaboration with NIST’s Italian counterpart, the Instituto Nazionale de Ricerca Metrologica (INRIM).

Light Modes

Generating light suitable for quantum mechanical applications such as quantum computing and quantum cryptography requires exquisite control over properties such as the frequency, polarization, timing, and direction of the light emitted. For instance, probing atoms with light involves matching the frequency of the light to the atoms’ natural resonance frequency often to within one part in a billion. Moreover, communicating with light means encoding information in the arrival time or frequency or spatial position of the light, so high-speed communication means using very closely- spaced arrival times for light pulses, and pinpoint knowledge of the light frequencies and positions of the arriving light in order to fit in as much information as possible.

When a light pulse contains a mixture of light photons with different frequencies (or polarizations, arrival times, emission angles, etc.) it is said to have multiple modes. In some cases, a single light source will naturally produce such multi-mode light, whereas in others multiple modes are a signature of the presence of additional, and generally unwanted, light sources in the system. Discriminating the different modes in a light field, especially a weak light field that has very few photons, can be extremely difficult as it requires very sensitive detection that can discriminate between modes that are very close together in frequency, space, time, etc.

For instance, to study a pair of entangled photons (created by shooting light into a special crystal where one photon is converted into a pair of secondary, related photons) detection efficiency is all important; and folded into that detection efficiency is a requirement that the arrival of each of the daughter photons be matched to the arrival of the other daughter photon. In addition to this temporal alignment, the spatial alignment of detectors, (each oriented at a specific angle respect to the beamline) must be exquisite. To correct for any type of less-than-perfect alignment, it is necessary to know how many different light modes are arriving at the detector.

Photon Number

The laws of quantum mechanics ensure that light always exhibits natural intensity fluctuations. Even from an ultra-stable laser, the number of photons arriving at a detector will vary randomly in time. By recording the number of photons in each pulse of light over a long time, however, the form of the fluctuations of a particular light field will become clear. In particular, we can learn the probability of generating 0, 1, 2, 3, etc. photons in each pulse.

The handy innovation in Migdall’s lab was to develop a method to use this set of probabilities to determine the modes in a very weak light field. This method is very useful because most light detectors that can see light at the level of a single photon cannot tell the exact frequency or position of the light, which makes determining the number of modes difficult for such fields.

The JQI-INRIM experiment used a detector “tree” that counts photon number. It did this by taking the incoming light pulse, using partial mirrors to divide the pulse into four, and then allowing these to enter four detectors set up to record individual photons. If the original pulse contained zero photons then none of the detectors will fire. If the pulse contained one photon, then one of the detectors will fire, and so on.

Elizabeth A. Goldschmidt, a JQI researcher and University of Maryland graduate student, is the first author on the research paper. “By looking at just the intensity fluctuations of a light field we have shown that we can learn about the underlying processes generating the light,” she said. “This is a novel use of higher-order photon-number statistics, which are becoming more and more accessible with modern photodetection.”

Goldschmidt believes that this method of counting photons and statistically analyzing the results as a way of understanding the modes present in light pulses will help in keeping tight control over light sources that emit single photons (where, for instance, you want to ensure that unwanted photons are not being produced). And those that emit pairs of entangled photons---where the quantum relation between the two photons is exactly right, such as in “heralding” experiments, where the detection of a photon in one detector serves as an announcement for the existence of a second, related, photon in a specially staged detector nearby.

Alan Migdall compares the photon counting approach to wine tasting. “Just as some experts can taste different flavors in a wine---a result of grapes coming from different parts of the Loire Valley---so we can tell apart various modes of light coming from a source.”

References:
[1] Elizabeth A. Goldschmidt, Fabrizio Piacentini, Ivano Ruo Berchera, Sergey V. Polyakov, Silke Peters, Stefan Kück, Giorgio Brida, Ivo P. Degiovanni, Alan Migdall, Marco Genovese, "Mode reconstruction of a light field by multi-photon statistics", Physical Review A, 88, (2013). Abstract.

Deterministic Quantum Teleportation

Christine A. Muschik (left) 
and 
Eugene S. Polzik (right)













Authors: Christine A. Muschik¹ and Eugene S. Polzik²

Affiliation:
¹ICFO - Institut de Ciències Fotòniques, Barcelona, Spain.
²Niels Bohr Institute and Danish Quantum Optics Center QUANTOP, Copenhagen University, Denmark.

Teleportation [1] is an essential element in quantum information science [2] and is intimately linked to and enabled by the distribution of entanglement between two locations.

The transmission of quantum information is a tricky business, since quantum systems are typically too fragile to be sent directly over large distances. Moreover, the no-cloning theorem [3] prevents one from realizing a faithful transmission by measuring a quantum state in one location and re-preparing it in another place.

Since teleportation provides a possibility to circumvent these problems, it is a key-ingredient in distributed quantum networks and lies at the heart of the practical realization of long distance quantum communication [4]. Apart from that, teleportation allows for universal quantum computing [5] if supplemented with local operations and suitable entangled resource states.

The first experimental realizations of teleportation protocols employed light as the carrier of quantum states [6]. The teleportation between matter systems is more challenging, but also particularly relevant. In contrast to photons, matter systems provide long-lived degrees of freedom which are required for the storage of quantum information. Matter systems are also needed for the efficient processing of quantum states and the extension to large quantum networks which involve multiple remote nodes.

The teleportation of quantum systems between matter systems has been demonstrated deterministically between ions that are held in the same trap [7]. However, for macroscopic distances, so far only probabilistic implementations were available, which yield random outcomes and succeed therefore only with a certain probability.

In our recent publication [8], we report on the realization of a novel protocol for teleportation over a macroscopic distance [9], which allows for the deterministic transfer of a quantum state, i.e. this scheme guarantees the successful teleportation of quantum states in every single attempt. This feature is important for technological applications and also opens up new possibilities for the teleportation of quantum dynamics [10].

The experiment is carried out using two atomic clouds, A and B, at room temperature which are contained in glass cells that are placed 0.5 meter apart [11]. Each atomic cloud contains approximately 10¹² Cesium atoms and the quantum state of each of these samples is encoded in the collective atomic spin state of the atomic cloud. A freely propagating laser field is used to entangle the two atomic samples and to teleport the spin state of cloud B to cloud A.

Figure 1: Deterministic quantum teleportation between two atomic clouds. A laser field is used to entangle the two samples and to teleport the spin state of cloud B to cloud A.

More specifically, the interaction between the laser field and a gas sample leads to entanglement between the atoms and the light. In our teleportation protocol, the laser field passes both samples, A and B, and is afterwards measured. This procedure first creates entanglement between the sample A and the light. The light then passes through the sample B and gets an imprint of its quantum state. The teleportation scheme is completed by performing a conditional feedback operation on cloud A: the spin state of sample A is displaced according to the measurement outcome using magnetic fields that are applied to the container of cloud A.

We used the deterministic character of the teleportation scheme to realize a “stroboscopic” teleportation, in which a sequence of rapidly changing spins states has been teleported between the two gas samples. This approach can be further extended to achieve a truly time-continuous teleportation, which paves the way toward the teleportation of quantum dynamics and the simulation of interactions between distant objects which cannot interact directly [10].

This work was done in collaboration with H. Krauter, D. Salart, J. M. Petersen, and H. Shen at the Niels Bohr Institute, Copenhagen, Denmark and T. Fernholz at the University of Nottingham, UK.

References:
[1] Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa, Asher Peres, and William K. Wootters, "Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels", Physical Review Letters, 70, 1895 (1993). Abstract.
[2] Michael A. Nielsen, Isaac L. Chuang, "Quantum Computation and Quantum Information". Cambridge University Press, Cambridge (2000).
[3] W.K. Wootters and W.H. Zurek, "A single quantum cannot be cloned", Nature, 299, 802 (1982). Abstract.
[4] H. J. Kimble, "The Quantum Internet", Nature, 453, 1023 (2008). Abstract.
[5] Daniel Gottesman and Isaac L. Chuang, "Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations", Nature, 402, 390 (1999). Abstract.
[6] Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter & Anton Zeilinger, "Experimental quantum teleportation", Nature, 390, 575 (1997). Abstract; A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, E. S. Polzik, "Unconditional Quantum Teleportation", Science 282, 706 (1998). Abstract.
[7] M. Riebe, H. Häffner, C.F. Roos, W. Hänsel, J. Benhelm, G.P.T. Lancaster, T.W. Körber, C. Becher, F. Schmidt-Kaler, D.F.V. James, R. Blatt, "Deterministic quantum teleportation with atoms", Nature 429, 734 (2004). Abstract  M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, D. J. Wineland, "Deterministic quantum teleportation of atomic qubits", Nature 429, 737 (2004). Abstract ; M Riebe, M Chwalla, J Benhelm, H Häffner, W Hänsel, C F Roos and R Blatt, "Quantum teleportation with atoms: quantum process tomography", New Journal of Physics, 9, 211 (2007). Abstract.
[8] H. Krauter, D. Salart, C. A. Muschik, J. M. Petersen, Heng Shen, T. Fernholz, and E. S. Polzik, "Deterministic quantum teleportation between distant atomic objects", Nature Physics, 9, 400 (2013). Abstract.
[9] Christine A. Muschik, "Quantum information processing with atoms and photons". Ph.D. thesis, Max-Planck Institute for Quantumoptics (2011). PDF file.
[10] Christine A. Muschik, Klemens Hammerer, Eugene S. Polzik, and Ignacio J. Cirac, "Quantum Teleportation of Dynamics and Effective Interactions between Remote Systems", Physical Review Letters, 111, 020501 (2013). Abstract.
[11] Klemens Hammerer, Anders S. SØrensen, and Eugene S. Polzik, "Quantum interface between light and atomic ensembles", Review of Modern Physics, 82, 1041 (2010). Abstract.

Quantum Computer Runs The Most Practically Useful Quantum Algorithm

Chao-Yang Lu (left) and Jian-Wei Pan (right) 












Authors: Chao-Yang Lu and Jian-Wei Pan

Affiliation: Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, China.

Over the past three decades, the promise of exponential speedup using quantum computing has spurred a world-wide interest in quantum information. To date, there are three most prominent quantum algorithms that can achieve this exponential speedup over classical computers. Historically, the first one is quantum simulation of complex systems proposed by Feynman in 1980s [1]. The second one is Shor’s algorithm (1994) for factoring large numbers [2] – a killer program to break the widely used RSA cryptographic codes.

Very recently, the third one came as a surprise. Harrow, Hassidim and Lloyd (2009) showed that quantum computers can offer an exponential speedup for solving systems of linear equations [3]. As the problem of solving linear equations is ubiquitous in virtually all areas of science and engineering (such as signal processing, economics, computer science, and physics), it would be fair to say that this might be the most practically useful quantum algorithm so far.

Fig.1: An optimized circuit with four qubits and four entangling gates for solving 2x2 systems of linear equations.

Demonstration of the powerful algorithms in a scalable quantum system has been considered as a milestone toward quantum computation. While the first two have been realized previously [4,5,6], the realization the new quantum algorithm had remained a challenge. Recently, we report the first demonstration of the quantum linear system algorithm, through testing the simplest meaningful instance: solving 2×2 linear equations on a photonic quantum computer [7], in parallel with Walther’s group who also presented results on arXiv [8]. To demonstrate the algorithm, we have implemented a quantum circuit (see Fig.1) with four quantum bits and four controlled logic gates, which is among the most sophisticated quantum circuit to date.

Fig.2: Experimental setup. It consists of (1) Qubit initialization, (2) Phase estimation, (3) R rotation, (4) Inverse phase estimation.

An illustration of our experimental set-up is shown in Fig.2. The four quantum bits are from four single photons generated using a nonlinear optical process called spontaneous parametric down-conversion (where a short, intense ultraviolet laser shines on a crystal and, with a tiny probability, an ultraviolet photon can split into two correlated infrared photons). The quantum information is encoded with the polarization of the single photons, which can be initialized and manipulated using wave plates, and readout using polarizers and single-photon detectors [9].

In the experiment, it is also necessary to implement four sets of photon-photon controlled logic gates – that is, the quantum state of a single photon controls that of another independent single photon. These gates are realized by optical networks consisting of polarizing beam splitters, half wave plates, Sagnac interferometer, and post-selection measurement.

Fig.3: Experimental results. For each input state, the experimentally measured (red bar) expectation values of the observables of the Pauli matrices are compared to the theoretically prediction (gray bar).

We have implemented the algorithm for various input vectors. We characterize the output by measuring the expectation values of the Pauli observables Z, X, and Y. Figure 3 shows both the ideal (gray bar) and experimentally obtained (red bar) expectation values for each observable. To compare how close our experimental results match ideal outcome, we compute the output state fidelities, which give 0.993(3), 0.825(13) and 0.836(16) for the three input vectors, respectively.

To solve more complicated linear and differential equations [10], we are trying to develop new techniques, including experimental manipulation of more photonic qubits, higher brightness multi-photon sources, and more efficient two-photon logic gates. So far, we have the ability to control up to eight individual single photons [11] and ten hyper-entangled quantum bits [12]. Creating a larger-scale circuit would involve more quantum bits. Two parallel pathways are being undertaken in our group. One is to climb up to ten-photon entanglement, and the other way is to exploit more degrees freedom of a single photon thus using it more efficiently. The near-future goal is to control 10 to 20 photonic quantum bits. The enhanced capability would allow us to test more complicated quantum algorithms.

References: 
[1] Richard P. Feynman, “Simulating physics with computers”. International Journal of Theoretical Physics, 21, 467 (1982). Abstract.
[2] P. Shor, “Algorithms for quantum computation: discrete logarithms and factoring” in Proc. 35th Annu. Symp. on the Foundations of Computer Science, edited by S. Goldwasser (IEEE Computer Society Press, Los Alamitos, California, 1994), p. 124–134. Abstract.
[3] Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, “Quantum algorithm for linear systems of equations”. Physical Review Letters, 
103, 150502 (2009). Abstract.
[4] Chao-Yang Lu, Daniel E. Browne, Tao Yang, and Jian-Wei Pan, “Demonstration of a compiled version of Shor’s quantum factoring algorithm using photonic qubits”. Physical Review Letters, 99, 250504 (2007). Abstract.
[5] B.P. Lanyon, T. Weinhold, N. Langford, M. Barbieri, D. James, A. Gilchrist, and A. White, “Experimental demonstration of a compiled version of Shor’s algorithm with quantum entanglement”. Physical Review Letters, 99, 250505 (2007). Abstract.
[6] Chao-Yang Lu, Wei-Bo Gao, Otfried Gühne, Xiao-Qi Zhou, Zeng-Bing Chen, Jian-Wei Pan, “Demonstrating Anyonic Fractional Statistics with a Six-Qubit Quantum Simulator”. Physical Review Letters, 102, 030502 (2009). Abstract.
[7] X.-D. Cai, C. Weedbrook, Z.-E. Su, M.-C. Chen, Mile Gu, M.-J. Zhu, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian-Wei Pan, “Experimental Quantum Computing to Solve Systems of Linear Equations”. Physical Review Letters, 110, 230501 (2013). Abstract.
[8] Stefanie Barz, Ivan Kassal, Martin Ringbauer, Yannick Ole Lipp, Borivoje Dakic, Alán Aspuru-Guzik, Philip Walther, “Solving systems of linear equations on a quantum computer”. arXiv:1302.1210v1. Abstract.
[9] Jian-Wei Pan, Zeng-Bing Chen, Chao-Yang Lu, Harald Weinfurter, Anton Zeilinger, Marek Żukowski, “Multiphoton entanglement and interferometry”. Review of Modern Physics, 84, 777 (2012). Abstract.
[10] Dominic W. Berry, “Quantum algorithms for solving linear differential equations”. arXiv:1010.2745. Abstract.
[11] Xing-Can Yao, Tian-Xiong Wang, Ping Xu, He Lu, Ge-Sheng Pan, Xiao-Hui Bao, Cheng-Zhi Peng, Chao-Yang Lu, Yu-Ao Chen, Jian-Wei Pan, “Observation of eight-photon entanglement”. Nature Photonics, 6, 225. Abstract.
[12] Wei-Bo Gao, Chao-Yang Lu, Xing-Can Yao, Ping Xu, Otfried Gühne, Alexander Goebel, Yu-Ao Chen, Cheng-Zhi Peng, Zeng-Bing Chen, Jian-Wei Pan, “Experimental demonstration of a hyper-entangled ten-qubit Schrödinger cat state”. Nature Physics, 6, 331 - 335 (2010). Abstract.

Quantum Information at Low Light

Alan Migdall of Joint Quantum Institute (JQI), a research partnership between University of Maryland (UMD) and the National Institute of Standards and Technology, USA [Photo Courtesy: NIST]

At low light, cats see better than humans. Electronic detectors do even better, but eventually they too become more prone to errors at very low light. The fundamental probabilistic nature of light makes it impossible to perfectly distinguish light from dark at very low intensity. However, by using quantum mechanics, one can find measurement schemes that can, at least for part of the time, perform measurements which are free of errors, even when the light intensity is very low.

The chief advantage of using such a dilute light beam is to reduce the power requirement. And this in turn means that encrypted data can be sent over longer distances, even up to distant satellites. Low power and high fidelity in reading data is especially important for transmitting and processing quantum information for secure communications and quantum computation. To facilitate this quantum capability you want a detector that sees well in the (almost) dark. Furthermore, in some secure communications applications it is preferable to occasionally avoid making a decision at all rather than to make an error.

A scheme demonstrated at the Joint Quantum Institute does exactly this. The JQI work, carried out in the lab of Alan Migdall and published in the journal Nature Communications, shows how one category of photo-detection system can make highly accurate readings of incoming information at the single-photon level by allowing the detector in some instances not to give a conclusive answer. Sometimes discretion is the better part of valor.

Quantum Morse Code:

Most digital data comes into your home or office in the form of pulsed light, usually encoding a stream of zeros and ones, the equivalent of the 19th century Morse code of dots and dashes. A more sophisticated data encoding scheme is one that uses not two but four states---0, 1, 2, 3 instead of the customary 0 and 1. This protocol can be conveniently implemented, for example, by having the four states correspond to four different phases of the light pulse. However, the phase states representing values of 0, 1, 2, and 3 have some overlap, and this produces ambiguity when you try to determine which state you have received. This overlap, which is inherent in the states, means that your measurement system sometimes gives you the wrong answer.

Migdall and his associates recently achieved the lowest error rate yet for a photodetector deciphering such a four-fold phase encoding of information in a light pulse. In fact, the error rate was some 4 times lower than what is possible with conventional measurement techniques. Such low error rates were achieved by implementing measurements for minimum error discrimination, or MED for short. This measurement is deterministic insofar as it always gives an answer, albeit with some chance of being wrong.

By contrast, one can instead perform measurements that are in principle error free by allowing some inconclusive results and giving up the deterministic character of the measurement outcomes. This probabilistic discrimination scheme, based on quantum mechanics, is called unambiguous state discrimination, or USD, and is beyond the capabilities of conventional measurement schemes.

Figure 1: Scheme for carrying out unambiguous state discrimination (USD). Inset (i) shows the four nonorthogonal symmetric coherent states with phases equal to ϕ = {0, π/2, π, 3π/2}. The state under measurement, |α_i>, has vertical (V) polarization, and the phase reference, |LO>, has horizontal (H) polarization. The pulse is distributed among four elimination stages using mirrors (M) and beam splitters (BS). Each elimination stage uses phase shifters (PS), a polarizer (Pol), and a single photon detector (SPD) to eliminate one possibility for the phase of the input state |α_i>.

In their latest result [1], JQI scientists implement a USD of such four-fold phase-encoded states by performing measurements able to eliminate all but one possible value for the input state---whether 0, 1, 2, or 3. This has the effect of determining the answer perfectly or not at all. Alan Migdall compares the earlier minimum error discrimination approach with the unambiguous state discrimination approach: “The former is a technique that always gets an answer albeit with some probability of being mistaken, while the latter is designed to get answers that in principle are never wrong, but at the expense of sometimes getting an answer that is the equivalent of ‘don't know.’ It’s as your mother said, ‘if you can’t say something good it is better to say nothing at all.’”

With USD you make a series of measurements that rule out each state in turn. Then by process of elimination you figure out which state it must be. However, sometimes you obtain an inconclusive result, for example when your measurements eliminate less than three of the four possibilities for the input state.

The Real World:

Measurement systems are not perfect, and ideal USD is not possible. Real-world imperfections produce some errors in the decoded information even in situations where USD appears to work smoothly. The JQI experiment, which implements USD for four quantum states encoded in pulses with average photon numbers of less than one photon, is robust against real-world imperfections. At the end, it performs with much lower errors than what could be achieved by any deterministic measurement, including MED. This advance will be useful for quantum information processing, quantum communications with many states, and fundamental studies of quantum measurements at the low-light level.

Reference:
[1] F.E. Becerra, J. Fan, A.L. Migdall, "Implementation of generalized quantum measurements for unambiguous discrimination of multiple non-orthogonal coherent states," Nature Communications, 4, 2028 (2013). Abstract.