nSQUID

IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 19, NO. 3, JUNE 2009 961

Progress Towards Reversible ComputingWith nSQUID Arrays

Jie Ren, Vasili K. Semenov, Yuri A. Polyakov, Dmitri V. Averin, and Jaw-Shen Tsai

Abstract—Circuits of nSQUIDs are expected to be able tooperate reversibly in both classical and quantum modes. Herewe present the first circuit that is fully operational classicallyat up to 5 GHz clock frequencies. The circuit contains two shiftregisters (i.e., cells that transfer the input data to the outputs) witha common clock ring. We estimate that the dissipated power isclose to the thermodynamic threshold of �� per switching.We also show that after a proper scaling of parameters, thenSQUID circuits should work similarly in the quantum mode. Inthis case, the unique advantage of nSQUID circuits is their abilityto transfer the quantum data along the chip and therefore formcomplex multi-qubit circuits bypassing the problem of controllablekeys. In the fully quantum regime, the nSQUID arrays shouldimplement directly the scheme of universal adiabatic quantumcomputation.

Index Terms—Long Josephson-junctions, quantum computing,reversible computing, superconductor electronic devices.

I. INTRODUCTION

T HE FIRST reversible circuits of nSQUIDs, i.e., dcSQUIDs with negative mutual inductance between

the arms of the SQUID loop, were introduced several yearsago [1] and were further improved later [2]. The two maingoals of our nSQUID effort were to demonstrate reversiblemedium-scale computations in the classical regime, and tosuggest an alternative approach to quantum computation (QC)with superconducting qubits. The principal advantages of thenSQUID circuits in the QC context originate from the twocircumstances. (1) These circuits do not require controllablekeys to switch on and off the interaction between the qubits.Such keys are critically important for other types of super-conducting qubits in order to organize them into larger QCstructures. (2) The quantum (qubits) and classical (support)parts of the nSQUID circuits can be naturally integrated intouniform structures that are actually scalable. In view of this,we strongly believe that the nSQUID approach will overcomethe initial skepticism and will prove very useful for the fur-ther development of superconducting quantum circuits. In

Manuscript received August 26, 2008. First published June 30, 2009urrentversion published July 10, 2009. This work was supported in part by the NationalSecurity Agency (NSA) Army Research Office (ARO) Contract W911NF-06-1-217, by CREST/JST.

J. Ren, V. K. Semenov, Y. A. Polyakov, and D. V. Averin are with theDepartment of Physics and Astronomy, Stony Brook University, Stony Brook,NY, 11794-3800 USA (e-mail: [email protected]; [email protected]; [email protected]).

J.-S. Tsai is with NEC, Nano Electronics Research Laboratories, 34 Miyuki-gaoka, Tsukuba 305-8501 Japan (e-mail: [email protected]).

Digital Object Identifier 10.1109/TASC.2009.2018250

particular, as shown in this work, the nSQUID circuits can beused as the natural basis for implementation of the universaladiabatic quantum computation (UAQC). In this approach,the computations are performed within the ground state of thecircuit Hamiltonian, with the energy gap between the groundand excited states of the Hamiltonian providing some degree ofprotection against decoherence.

The other important aspect of the nSQUID circuits is thatthey can be usefully operated in the classical mode [1]. In thiscapacity, they are positioned as the physically and logically re-versible digital circuits [3]–[5] with extremely low energy dissi-pation that can approach or even cross the psychologically im-portant thermodynamic threshold of per logic oper-ation. This energy dissipation is about 4 orders of magnitudesmaller that the energy dissipation in RSFQ circuits.

This is our third paper devoted to physically and logicallyreversible nSQUID circuits. We start by reminding the basicprincipal of their operation, then present and discuss the resultsof measurement of the first fully operational circuit, and con-clude with a discussion of prospective QC applications of thenSQUIDs, in particular to implementation of the UAQC algo-rithms.

II. COMPARISON OF RSFQ AND nSQUID CIRCUITS

As was already mentioned in the introduction, nSQUID cir-cuits dissipate dramatically less power than their RSFQ coun-terparts. To see this, one can look at the “primitive” RSFQ cellshown in Fig. 1: two similar pieces of Josephson TransmissionLines (JTLs) that transfer the SFQ pulses (or vortices) whichrepresent clock and data signals from JTL inputs to their out-puts. Josephson junctions in JTLs are critically dumped by shuntresistors and relatively large bias currents are applied to thejunctions to compensate for the viscous friction between vor-tices and shunt resistors. The bias currents for many junctionsare delivered via a single power line and are distributed betweenthe junctions using the corresponding bias resistors . Most ofthe energy is dissipated in this case in the bias resistors ratherthan in the “active” circuitry. This drawback is practically un-avoidable for two reasons. First, the required bias voltageis too low to keep it constant directly by means of semicon-ductor electronics. As a result, the constant current rather thanvoltage is supplied by the power source. The second factor isthe Josephson voltage-to-frequency relationship which impliesthat the average voltage across the JTL (shown as in Fig. 1)fluctuates following the variable rate of flow of the data vortices.The distribution of the total bias current between the clock anddata lines depends then on the data pattern. This parasitic effectis reduced to an acceptable level if the effective value of bias

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962 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 19, NO. 3, JUNE 2009

Fig. 1. Data flow in a typical RSFQ cell (Clock and Data JTLs). For discussion,see main text.

Fig. 2. nSQUID circuits: (a) schematics and (b) layout of an individualnSQUID. (c) Data flow diagram in a string of nSQUIDs coupled through pairsof inductive strips with negative mutual inductance between them.

voltage is significantly higher than the variations of the voltage.

A string of nSQUIDs (Fig. 2) is similar to a JTL but withJosephson junctions replaced by nSQUIDs (Figs. 2(a) and 2(b)).Each nSQUID [1], [2] is a 2-junction SQUID with a negativemutual inductance between its inductive arms. This negativemutual inductance resolves the two conflicting requirements forthe two degrees of freedom of the system [1], [2]. In a commonmode, which represents common dynamics of the two SQUIDjunctions, one would like to keep the low effective inductance

, and therefore, simple dynamics of this mode gov-erned by the current bias of the junctions via the Clock line. Forthe differential mode, which represents the current circulatingalong the SQUID arms, one needs to provide a larger effectiveinductance , so that there are two stable state possiblein this degree of freedom.

The low inductance of the common mode of thestrings of nSQUIDs connected as in Fig. 2(c), implies that thestrings support propagation of vortices along the strings. Theproperties of these vortices are close to those of the vortices in

long Josephson junctions. The main difference between the twosituations is that the nSQUIDs located near the current centersof the moving vortices experience approximately , i.e. ,magnetic (clock) bias and therefore find themselves in one ofthe two stable states that can be distinguished by the sign ofthe current circulating along the nSQUID [1], [2]. Because ofthe relatively strong magnetic coupling of the nearest-neighbornSQUIDs in the string (Fig. 2(c)), all nSQUIDs which belong toone vortex share the same logic state. This state can be naturallyused to carry one bit (“0” and “1”) of logic (digital) data.

This data presentation has two advantages. First, the averagevoltage on the clock line is proportional to the frequency ofthe clock vortices and is independent of the data pattern. This in-dependence of the data flow makes it possible to dramatically re-duce the bias resistors and the bias voltage, if the biasing schemeshown in Fig. 1 is used. However, as we will discuss later, thetotal bias current is so low that it can be applied directly to theclock line. The second advantage is that the data domain is au-tomatically “synchronized” with the clock vortex, and the timejitter between the clock and the data is essentially suppressed.The practical consequence of this is that the critical currents ofthe Josephson junctions can be reduced. In addition to all this,the dynamics of the nSQUID string is quite simple. In fact, itis nothing more than a motion with constant speed of a clockvortex coupled with data domain. In this case, the shunt resis-tors can be completely eliminated, as one would expect for areversible circuit.

III. OPTIMIZATION OF EXPERIMENTAL CIRCUIT FOR MINIMAL

ENERGY DISSIPATION

In the string of nSQUIDs (Fig. 2(c)), we have eliminated thedissipation of energy in the bias and shunt resistors. There isstill, however, a considerable energy flow associated with theclock vortices themselves. To avoid dissipating this energy, thevortices, together with their energy, can be “recycled”, e.g., byconnecting the clock lines of the two shift registers to form aring as shown in Fig. 3. In the simplest regime of operation, thering contains then the two vortices ( in Fig. 3(a)), oneper each shift register, which propagate in opposite directions.Dynamics of vortices in this structure is similar to the vortexmotion in unshunted ring-shaped Josephson junctions. Ideally,such ring junctions have zero critical currents (i.e., applicationof any non-vanishing bias current leads to vortex motion), andthe principal energy-dissipation channel for slowly moving vor-tices is the sub-gap junction leakage current, roughly propor-tional to the vortex speed.

Therefore, in the ideal regime, this model of vortex motion isindeed characterized by zero energy dissipation. However, anyimperfections, e.g., the discrete structure the junction, or non-uniformity of the critical current density immediately inducesome finite critical current and additional dissipation mecha-nisms. In the regime of slowly moving vortices, the extra energydissipation can be described in terms of variations of the speedof moving vortices. To suppress these variations, and thereforethe variations of the voltage across the array, as strongly as pos-sible, we need the source of the bias voltage with minimuminternal impedance. Such a voltage source has been temporallyimplemented as a resistor with low resistance (Fig. 4). The

REN et al.: PROGRESS TOWARDS REVERSIBLE COMPUTING WITH nSQUID ARRAYS 963

Fig. 3. (a) Equivalent circuit and (b) the microphotograph of the two shift reg-isters with a common clock ring. The length of one cell is 180 ��, the lengthof the ring is 1410 ��. Only 6 of 8 cells are shown in part (a).

Fig. 4. Diagram of the setup for measurement of the energy dissipation in theshift register structure shown in Fig. 3. The bias current �� flowing through thebias resistor�� creates the voltage bias � �� for the shift registers that is appliedthrough the resistor � shunting the inductance of the current-measuring partof the circuit at large frequencies. The current through the shift registers containsboth the dc and ac parts �� and � . The double-sided arrow indicates parasiticcoupling between the bias current and the current-measuring SQIF.

required dc voltage is obtained by applying large currentto this resistor. Since only one such voltage source per circuitis needed, and its structure is independent of the circuit com-plexity, we regard it as external to the circuit and exclude theenergy dissipation in the voltage source from our energy bal-ance. Nevertheless, as one of the next steps, we plan to imple-ment more advances “capacitive” voltage source that would notdissipate energy.

Full diagram of the setup we used for measurements of the en-ergy dissipation is shown in Fig. 4. The Josephson voltage-to-frequency relationship implies that the measurement of the en-ergy dissipated in an nSQUID circuit is reduced to the measure-ment of dc current flowing through the circuit [2]. Indeed, theenergy dissipated during each logic operation can be obtainedby multiplying the dissipated power

(1)

by the period of the operation:

(2)

Combined with the Josephson voltage-to-frequency relation-ship:

(3)

this equation shows that the dissipation energy is determinedonly by the current :

(4)

As a result, comparison of the energy dissipation with its ther-modynamic threshold is equivalent to the com-parison of the dc current with the “threshold current”

(5)

At liquid helium temperature of 4 K, the threshold current is.

In our chosen measurement setup (Fig. 4), the current is mea-sured using a compensation technique, with a SQIF (see, e.g.,[6]) as the null detector. There are two limitations to such a mea-surement scheme. One is technical: a finite parasitic couplingbetween the bias current and the SQIF. Another, more fun-damental limitation is related to the non-vanishing ac compo-nent of the current through the measured shift register cir-cuit. This component is created by the Josephson oscillationsin the circuit which acquires finite critical current becauseof the discreteness of the structure and/or other possible im-perfections. As usual, Josephson oscillations are characterizedby, in general complex, relations between the power dissipationat different frequencies [7]. The limitation of our measurementsetup is associated with the conversion of the zero frequencypower to ac power at Josephson frequency . Al-though this energy is dissipated physically in the resistor ofthe voltage source, it is registered by the SQIF as correctionto the dc current through the shift register circuit. In the rel-evant overdamped regime, when the circuit capacitance canbe neglected, can be estimated by modeling the circuit as oneJosephson junctions with critical current :

(6)

This equation shows that this correction to the current decreasesrapidly with increasing bias current . The need to makesmall, so that it does not preclude the measurement of energydissipated in the circuit, was the main reason for making the


Fig. 5. Numerically simulated behavior of the circuit shown in Fig. 3.

resistance in our circuit very low (0.0017 Ohm for chipRC07A, 0.0057 Ohm for chip RC05). Small leads to largevalue of at fixed bias voltage that determines the circuitoperation frequency. As soon as there is no need to prove thelow energy dissipation anymore, or a better measurement tech-nique, e.g. based on a capacitive voltage source, is implemented,the resistance can be dramatically increased.

We have simulated the actual time dependence of the currentin the circuit of two shift registers (Fig. 3) using our conven-tional tool PSCAN. Fig. 5 shows simulation results for parame-ters close to their experimental values. It shows that the ampli-tude of current flowing in the individual nSQUIDs is quite large,about 10 , i.e. its amplitude is about 2 times larger than thecritical current of a single Josephson junction. The current indifferent nSQUIDs, however, compensate each other, so that thetotal current through the circuit does not scale with the numberof nSQUIDs (Fig. 5), and roughly has the same amplitude asthe current in one nSQUID. Moreover, its average value is stillmuch lower than its ac amplitude.

IV. CIRCUIT FABRICATION

General characteristic feature of our circuits (an exampleshown in Fig. 6) is their complexity. Such level of complexitymeans that the circuits can be fabricated only at the dedicatedmicroelectronics facilities rather than university labs. Our de-signs were optimized for fabrication at two of the best facilitiesspecializing in superconductor circuits: HYPRES, Inc. andISTEC. Together, they cover a wide range of junction criticalcurrent densities: from 30 (HYPRES), to 350(ISTEC) and 1000 (HYPRES). Below we will discussthe results obtained with 30 and 1000 criticalcurrent densities. Because of the new character of the reversiblenSQUID circuits, the development process has required manyiterations to resolve design and (some of the) measurementproblems. The sixth revision of the shift register circuit showedan appropriate functionality, with only the seventh and eighthrevisions demonstrating a reasonable level of quantitativeagreement with our numerical estimates.

V. MEASUREMENTS

Our general discussion of the vortex dynamics in strings ofnSQUID was limited above to regular, i.e. infinite, strings. An

Fig. 6. Example of the “floor plan” of the chip. (Shown area is about 4 mm �4 mm). The chip was fabricated at HYPRES, Inc., wafer number kl1094.

Fig. 7. The universal read/write cell that terminates both ends of the twoshift registers in our ring structure: (a) equivalent circuit, (b) micrograph, and(c) measured output characteristics of the readout SQUID.

important addition one needs to make in the case of practicalcircuits is that all finite strings of nSQUIDs should be prop-erly terminated. For the nSQUID string to play a role of re-versible computational device, the cells at the ends of the stringshould perform the functions of the data input/output, and dothis without destroying the circuit reversibility. A “geometric”aspect of this reversibility is that the direction of the data flowin the circuit should also be in principle easily reversible. It isappropriate then for the end cells to be able to perform boththe input and the output function. An example of such a cellused in our shift registers is shown in Fig. 7. The data readoutis performed by the SQUID shown in the lower part of the cell


Fig. 8. Digitization and propagation of the digitized data along the shift reg-ister. Analog input magnetic flux is created by current Iin, digitized output isextracted at the other end of the register by the readout SQUID (for notations,see Fig. 7(a)).

(Fig. 7(a)), which measures the differential magnetic flux rep-resenting logic data stored in the corresponding nSQUID. Thetwo SQUIDs are only weakly coupled (the corresponding cou-pling coefficient is ) to suppress their undesirable in-teraction. At the same time, this coupling is sufficient for mea-surements of the flux state due to high sensitivity of the readoutSQUID (see Fig. 7(c)).

The data are written into the nSQUID strings by applyingthe differential dc magnetic bias created by the current(Fig. 7(a)). The nSQUID strings act as the shift register,i.e. the data are written into the nSQUIDs at one end of thestring, transported along it by the propagating vortex, andare measured at its other end. Fig. 8 shows typical results ofsuch measurements which illustrate the digitization effect:at lower values of the analog input signal, the digital outputshows constant (digital) output corresponding to a negativedifferential state of the nSQUIDs (logical “0”), while at higherinput signal, the differential state is positive (logical “1”). Theoverall process, viewed as “calculation”, is rather simple. Itinvolves 3 primitive functions: writing bits of data into a shiftregister, propagating them by about 2 mm distance, and readingthem out. The novel feature of this calculation is extremely lowenergy dissipation, comparable to the basic thermodynamicscale of energy dissipation.

The measurement procedure used to measure the energy dis-sipation in the shift registers was discussed above—see Fig. 4and related text. Although in the tested structures we have so farnot made this measurement to work properly, we briefly presentsthe results. Fig. 9 shows typical measured dependence of cur-rent , which represents the dissipated energy, on the biascurrent . It shows that when is below the critical current ofour circuit, all the bias current flows via the shift registers. How-ever, after crosses its critical value (4 and 6.6 for the twosamples), current through the circuit and, correspondingly

, decreases. Dashed lines show this dependence for thesimple model discussed earlier. The fit, especially for the sample

Fig. 9. Measurements (solid lines) and estimates (dashed lines) of the energydissipated in the nSQUID shift registers. The sample RC07 (wafer No. kl1107)is fabricated using 30 �� technology; RC05 (wafer No. kl1083)—with1000 �� technology. For discussion, see text.

RC07, is poor. The main reason for this is parasitic coupling be-tween the bias current and the measuring SQIF, which wastoo large in this sample. As a result, at the moment we can onlyroughly estimate the dissipated energy.

The principal dissipation sources are the sub-gap resistancesof the unshunted Josephson junctions in nSQUIDs. The totalcritical current of junctions in one shift register (about 0.08 mA),and the sub-gap voltage (about 70 mV) reported by HYPRESgive the estimate of the cumulative sub-gap resistance of theshift register of about 875 Ohm. Current via this resistance isproportional to the applied voltage and, therefore, the operationfrequency given in Fig. 8. It remains below the thermodynamicthreshold current (5) in the whole frequency range of correctoperation of the shift registers in Fig. 8.

As discussed in Section III, another component of the energydissipation is the “detection” of the Josephson oscillations in theresistance of the voltage source. The estimate (6) shows thatthe current measure of the magnitude of this energy dissipa-tion equals the threshold at frequency close to 0.2 GHz anddecreases as with increasing frequency. These estimatesmean that within the whole range of frequencies where our cir-cuit is operational: it should have energy dissipation below thethermodynamic threshold. Because of the measurement prob-lems discussed above, however, we could not support these esti-mates by direct measurement, and can only claim that the energydissipated in our shift registers is close to the thermodynamicthreshold.

VI. QUANTUM MODE OF OPERATION

Earlier [2] we introduced the “Flying Qubit Logic” (FQL)that could be built using the nSQUID arrays. It has an important,and probably unique among the solid-state quits, advantage: thepossibility to transfer quantum data along the circuit and, inthis way, to bypass the problem of controllable keys. This fea-ture is significant, since it addresses one of the main difficultiesfacing the gate-model quantum computation, the need to pre-cisely time the qubit-qubit interactions. Indeed, any two-qubitquantum logic operation can be viewed as a controlled rotationin the appropriate part of the Hilbert space. The rotation angle


depends on the relevant interaction energy of the two qubitsand the time duration of the operation:

(7)

The main example of this is the controlled-NOT gate, whichimplements the rotation by , i.e. inversion ofthe state of the target qubit in the computational basis. To im-plement this rotation, one needs to switch the interaction on forprecisely defined time interval such that

(8)

This equation gives the relation between the coupling energyand the time necessary to complete the quantum rotation. It illus-trates the requirement of precise time-control of the qubit-qubitinteraction that exists in the gate-model quantum computations.From practical perspective, the problem of development of con-trollable keys capable of switching the interaction on and offwith the required accuracy is no less difficult that the much moreexhaustively discussed problem of making the qubit decoher-ence sufficiently weak.

The nSQUID circuits bypass this problem of precise timecontrol of the interaction by effectively replacing time evolu-tion of the qubit states with propagation along the circuit. In thenSQUID arrays, the interaction of qubits necessary for the rota-tion and resulting logic operation is provided by coupling of thetwo strings of nSQUIDs, e.g., as shown in Fig. 10. Interactionenergy is estimated then as , where and

are the differential fluxes in the first and the second nSQUIDstrings, which have rather limited ranges, say, from to

, while any desirable value of can be implementedrather easily. Time of interaction is inversely proportional tothe speed of moving qubits. Of course, the realistic time depen-dence of the interaction will not have the rectangular profile,but it still can be presented as a pulse (Fig. 10(b)), with onlythe total integral of this pulse over time relevant for the magni-tude of the state rotation. The controlled-NOT gate used as anexample above would require one nSQUID string to affect thejunction critical current and therefore the tunnel splitting in theother nSQUID string through the coupling inductance.

As usual, the controlled-NOT gate discussed above, com-bined with the similarly implemented rotations of individualqubits, is in principle all that is needed to build an arbitrary com-plex multi-qubit circuit of nSQUIDs. Fig. 11 shows the sketchof the general structure of such a circuit, where the strings ofnSQUIDs are shown by solid lines and the two-input gates areshown by the double-sided arrows. As with other type of qubits,this structure is also convenient for a simple set of initial exper-iments. Its main new feature would be the ability to “adjust” theinteraction parameters by changing the speed of flying qubits,e.g. by changing the voltage bias of the clock lines.

The quantum operation of the nSQUID arrays describedabove assumes that the two modes of the nSQUIDs, “internal”differential mode, and “external” common mode exhibit dif-ferent types of the dynamics. The differential mode whichencodes the digital data is quantum coherent, and therefore,each internal state of the fluxon in the nSQUID string carries

Fig. 10. A two-input quantum logic gate. (a)—equivalent circuit and notation,(b)—dependence of the interaction energy of the two moving qubits on time.

Fig. 11. General structure of nSQUID network operating in quantum mode.Arrows represent gates shown in Fig. 10.

one qubit of quantum information. By contrast, the dynamicsof the collective mode which is employed to transfer thefluxons along the array is classical, i.e. each fluxon is in definiteposition in the array and is driven along it by the voltageapplied to the clock line. In principle, similarly to the fluxonsin long Josephson junctions [8], the junction parameters for thenSQUIDs (most importantly, the area) can be chosen in such away—see, e.g., [9], that the fluxons in nSQUID array propagateballistically, and this ballistic propagation can be quantumcoherent if the decoherence is sufficiently weak. An importantfeature of this regime is that the arrangement of several parallelnSQUID arrays with the structure of the type shown in Fig. 11,realizes the general scheme of universal adiabatic quantumcomputation (UAQC).

UAQC [10], [11] is the approach to quantum computationwhich is equivalent to the more standard gate-model quantumcomputation. Both historically and logically this scheme ofquantum computation is closest to thermodynamically re-versible classical algorithms. Its overall structure includes


computational qubits and the clock register, and is representedby the “Feynman Hamiltonian” [12], [13]:

(9)

Here is the unitary transformation performed on the com-putational qubits at the th step of the computation, and arethe states of the “clock” register, the qubits in which encodethe transformation number and therefore keep track of the or-dering of . In the nSQUID arrays, the role of the clock registeris played by the common modes of the nSQUIDs in all stringsof the array, while the computational qubits are represented bythe fluxon differential modes.

When the common mode dynamics of the nSQUID is clas-sical, i.e. when the circuit either realizes the classical reversiblecomputation, or FQL, the main purpose of the common modesof the nSQUIDs is to synchronize the fluxon dynamics in dif-ferent strings (by appropriate inductive coupling between thestrings) as needed for the operation of the different logic el-ements in the nSQUID structure in Fig. 11. For all the cou-plings shown in Fig. 11 to perform logic functions as describedat the beginning of this section, the fluxons in the two coupledlines should be traversing the interaction region during at thesame time. This condition is satisfied most straightforwardly,if the fluxons in all nSQUID strings propagate together, as acommon information-carrying wave. This type of the synchro-nized fluxon dynamics is needed regardless of whether the infor-mation is represented in this wave by classical or quantum-co-herent states of the differential nSQUID modes, demonstratingdeep similarity between the information propagation patterns inthe classical and quantum reversible computations.

In the UAQC regime, when the common mode dynamics isalso quantum, it acquires additional purpose besides synchro-nization of the information propagation through the circuit. Itensures that while the wavefunction of the computational qubitsencoded by the differential modes of the nSQUIDs within thepropagating fluxons represents some meaningful computation,the overall state of the fluxons remains the ground state of thenSQUID array (Fig. 11) as a quantum system. The main advan-tage of performing quantum computation in this way is somedegree of protection against decoherence afforded by the en-ergy gap separating the ground from excited states of thearray. At the minimum, such a gap makes it possible to mean-ingfully operate the circuit for times much longer that the co-herence time of individual qubits [14]. An open problem of

this approach is the fact that the energy gap typically de-creases rapidly with the size of the quantum algorithm. As onecan see explicitly for the nSQUID realization of the UAQC ap-proach, the gap is produced physically by the quantizationof ballistic motion of a massive object, synchronized wave offluxons in coupled nSQUID strings which are moving ballisti-cally along the array. It is obviously quite a challenging taskto make the size-quantization gap associated with this motionsufficiently large. Still, protecting quantum states with the en-ergy gap maybe the best approach to solving the decoherenceproblem in future large-scale quantum computation.

ACKNOWLEDGMENT

The authors thank S. Tolpygo, O. A. Mukhanov, S. Han, M.Hidaka, S. Yorozu, and K.K. Likharev for useful discussions.

REFERENCES

[1] V. K. Semenov, G. V. Danilov, and D. V. Averin, “Negative-inductanceSQUID as the basic element of reversible Josephson-junction circuits,”IEEE Trans. Appl. Supecond., vol. 13, no. 2, pp. 938–943, 2003.

[2] V. K. Semenov, G. V. Danilov, and D. V. Averin, “Classical andquantum operation modes of the reversible Josephson-junction logiccircuits,” IEEE Trans. Appl. Supecond., vol. 17, no. 2, pp. 455–461,2007.

[3] C. Bennett, “Logical reversibility of computation,” IBM J. Res. Devel.,vol. 17, pp. 525–532, 1973.

[4] R. Landauer, “Uncertainty principle and minimal energy dissipation inthe computer,” Int. J. Theor. Phys., vol. 21, pp. 283–297, 1982.

[5] K. K. Likharev, “Classical and quantum limitations on energy con-sumption in computation,” Int. J. Theor. Phys., vol. 21, pp. 311–326,1982.

[6] J. Beyer and D. Drung, “A SQUID series array dc current sensor,” Su-percond. Sci. Technol., vol. 21, 2008, #095012.

[7] P. Russer, “General energy relations for Josephson junctions,” Proc. ofIEEE, vol. 59, p. 282, 1971, #2.

[8] A. Wallraff, A. Lukashenko, J. Lisenfeld, A. Kemp, M. V. Fistul, Y.Koval, and A. V. Ustinov, “Quantum dynamics of a single vortex,”Nature, vol. 425, pp. 155–158, 2003.

[9] D. V. Averin, K. Rabenstein, and V. K. Semenov, “Rapid ballisticreadout for flux qubits,” Phys. Rev. B, vol. 73, p. 094504, 2006.

[10] D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, and O.Regev, “Adiabatic quantum computation is equivalent to standardquantum computation,” SIAM J. Comput., vol. 37, pp. 166–194, 2007.

[11] A. Mizel, D. A. Lidar, and M. Mitchell, “Simple proof of equivalencebetween adiabatic quantum computation and the circuit model,” Phys.Rev. Lett., vol. 99, 2007, #070502.

[12] J. Kempe, A. Kitaev, and O. Regev, “The complexity of the localHamiltonian problem,” SIAM J. Comput., vol. 35, pp. 1070–1097,2006.

[13] S. Lloyd, Robustness of Adiabatic Quantum Computing arXiv:0805.2757.

[14] M. H. S. Amin, C. J. S. Truncik, and D. V. Averin, Role of theSingle Qubit Decoherence Time in Adiabatic Quantum ComputationarXiv:0803.1196.

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