degrees of freedom into correlation with a quantum system is sufficient to destroy ... suggested for the qubits of a quantum computer are of atomic or near-atomic ...
Prospects for Quantum Coherent Computation Using Superconducting Electronics Mark F. Bocko, Andrea M. Herra and Marc J. Feldman University of Rochester, Department of Electrical Engineering, Rochester, New York, USA
Abstract—We discuss the prospects and challenges for implementing a quantum computer using superconducting electronics. It appears that Josephson junction devices operating at milli-Kelvin temperatures can achieve a quantum dephasing time of milliseconds, allowing quantum coherent computations of 1010 or more steps. This figure of merit is comparable to that of atomic systems currently being studied for quantum computation.
I. INTRODUCTION In quantum coherent computation information is coded not just as “1” and “0” but also as coherent superpositions of the “1” and “0” states of a quantum mechanical two state system. Recent experiments from atomic and optical physics have demonstrated the creation and manipulation of such quantum mechanical bits, so-called ‘qubits’ [1]-[3], and consideration is being given to the prospects for constructing simple quantum computers. In this paper we will discuss the prospects for a superconducting electronics implementation of quantum computation. The great difficulty of quantum coherent computation is to prevent the quantum modes that are used to encode information from interacting with any other degrees of freedom except as prescribed by the computation. Otherwise the quantum mechanical coherence will be lost. Quantum coherence is extraordinarily fragile and may be easily destroyed by interactions with the environment, even if there is no exchange of energy between the quantum mode and the environment! Simply bringing an environment with many degrees of freedom into correlation with a quantum system is sufficient to destroy a quantum superposition state [4]. Most if not all of the physical systems that have been suggested for the qubits of a quantum computer are of atomic or near-atomic dimensions. Macroscopic objects are often dismissed as being unsuitable (for instance [5]). The problem is that macroscopic objects generally have a large number of internal degrees of freedom, elastic vibrations, conduction electron excitations, etc., and so it is very difficult to prevent the coding mode, the qubit, from interacting with other internal modes of the device. Therefore, the time over which quantum coherence can be maintained is very short for most macroscopic objects. Superconductivity is clearly a macroscopic quantum effect. In this context, however, the quantum nature of superconductivity is secondary to another aspect which is far more important for the realization of a physical macroscopic quantum computer; this is the characteristic "quantum purity" of superconductors at very low temperatures. Far below the transition temperature, superconductors have extremely low entropy -- note the vanishing specific heat -- which means that there are very few available internal modes. Most of the
Manuscript received August 25, 1996. Also with the Dept. of Phys. and Astron., University of Rochester.
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dissipative mechanisms which normally operate in macroscopic systems are eliminated, and therefore electromagnetic collective excitations such as the Josephson plasma resonance are much more weakly damped than in normal metals. It is this property which makes superconductivity a candidate for quantum coherent computation -perhaps the only macroscopic candidate. Macroscopic superconductor-based systems for quantum computation have one very strong advantage compared to atomic-scale systems; superconductor circuit technology is a well-established art. If simple quantum gates can be devised and demonstrated, it will be relatively straightforward to scale to larger, more complex circuits using established integrated circuit technology. Such scaling is inconceivable for many of the current atomic-scale quantum computation schemes. It will likely take several years and require various technological advances before the simplest superconducting quantum logic gate can be built, but after that there are fewer impediments to the realization of large scale quantum computation. Only then may quantum computation conceivably compete with conventional computers, for instance using Shor’s algorithm to find the prime factors of large numbers [6]. The futility of searching for the prime factors of large numbers is due to the exponential dependence of the computational steps required to complete the task on the number of bits in the number; this provides the basis for the widely used RSA encryption scheme. In a broadly publicized event, 5000 MIPS-years of distributed computing resources were devoted over a period of eight months to factoring a 129 decimal digit number in answer to the RSA factoring challenge [7]. It is a simple matter to generate a number of 300 decimal digits that would take many times the age of the universe to factor using current algorithms and computing resources. In contrast, Shor’s quantum factoring algorithm may find the prime factors of a number in a time that is only a polynomial function of the number of bits. The vast increase in efficiency is afforded by the intrinsic parallelism which allows a quantum computer to explore all combinations of “1”’s and “0”’s simultaneously. II. PHYSICAL REALIZATION A physical quantum computer requires the following three elements: i. a two-state quantum system, a qubit, ii. a means of preparing the initial quantum state of the qubits, and iii. a way to implement logical interactions of the qubits. A prevalent model of a qubit is the spin-1/2 particle for which the single degree of freedom, the spin projection in an externally applied magnetic field, can take only two values. This is an intrinsic two-state system and it is common to designate the spin-up state as |1〉 and the spin-down as |0〉. Transitions between the |1〉 and |0〉 states may be induced by applying a pulse of electromagnetic radiation tuned to the energy difference of the states and with sufficient duration to cause a transition; a so-called π pulse. For a pulse of half that duration, a so-called π/2 pulse, the spin qubit will change to the superposition state [|0〉 - |1〉]/√2. Finally,
logical operations involving the spin qubits could be arranged by taking advantage of the local interactions among neighboring qubits [8]. In the following, we discuss possible superconducting implementations of the required elements of a quantum computer. A. The Superconducting Qubit A qubit is a quantum mechanical system with a state that is effectively restricted to a two-dimensional Hilbert space. In addition to the spin-1/2 system another possibility is the double potential well, a system with a continuous degree of freedom associated with a potential energy function with two symmetric minima separated by an energy barrier. It is natural to designate the state localized in one of the wells as |1〉 and in the other well as |0〉. The rf SQUID is an example of this type of two-state system; in fact, the Hamiltonians of the spin-1/2 particle in a magnetic field and of the isolated rf SQUID are completely equivalent [9]. The rf SQUID consists of a single Josephson tunnel junction with critical current Ic shunted by an inductor L. If 1 < 2πLIc /Φ 0 < 5π/2 and we externally apply to the loop a magnetic flux equal to one-half of the fundamental flux quantum Φ 0 , then a symmetric two state quantum system is created. The two states correspond to the loop containing either zero or one flux quantum, i.e., there is a supercurrent circulating in the SQUID in one or the other direction. A possible variation on the rf SQUID qubit is based on the quantum flux parametron (QFP) [10]. The QFP is essentially a symmetric pair of back-to-back rf SQUIDs sharing a common inductor. The parameters are chosen so that a single flux quantum is induced in the entire loop but the shared inductor forces the flux to choose one side or the other. A small offset current through the shared inductor breaks the symmetry and determine which side the flux will choose. Circulating currents in the QFP may then be inductively tapped to provide the offset currents for other QFPs. The energy versus flux diagram for the QFP is a double well, like that of the rf SQUID. A flux quantum in the right loop may represent a |1〉 and in the left loop a |0〉. The QFP symmetry assures that the quantum mechanical basis states are the even and odd combination of these, so a coherent mixture of |1〉 and |0〉 results for zero offset current. The QFP based qubit may have an advantage over the rf SQUID since its symmetry makes it less subject to fabrication-induced parameter variations, which are likely to affect both Josephson junctions equally. B. Preparation of the Qubit Initial State Devising a method to establish a superposition state in a superconducting qubit is a fundamental requirement. A good deal of attention has been paid to quantum superpositions of the two macroscopically distinct flux states in an rf SQUID in the context of macroscopic quantum coherence [11]. However, in that context one is concerned with the oscillation between two degenerate minima in the double-well potential. Initially, the wavefunction can be localized in one of the minima by making a measurement of flux, with the outcome 0 or 1 flux quantum, then the SQUID would be allowed to evolve freely. At certain phases of the oscillation the SQUID will be in a superposition of the two localized states. However, one must find a way to “freeze” the SQUID in the superposition state. A possible alternative method for placing a SQUID qubit in a superposition state may be to perform an energy measurement on the SQUID [12]. The
lowest two energy eigenfunctions of the rf SQUID are the symmetric and antisymmetric superpositions of wave functions peaked at the locations of the two wells. Finally, one may also control the SQUID bias current to manipulate the quantum state in analogy to the π and π/2 pulses used for particle spin qubits. The QFP may be easier to establish in a quantum superposition state since classically there is an equal probability of finding the flux quantum in either side of the QFP for zero offset current. In any case, finding a technique for establishing initial states in the superconducting qubit is clearly a fundamental problem to be solved. C. Logical Interactions of Qubits The next requirement is to develop a means of allowing the qubits to interact to achieve a given set of logical operations. The set of gates that may best achieve a given functionality is application and technology dependent. Here we suggest only one possibility. The COPY, NOT, and AND gates are sufficient to accomplish a general logical operation and in this section we will describe how rf SQUID-based circuits may be constructed to achieve these three basic functions. Nearly 20 years ago Likharev [13] proposed a device he called the parametric quantron which was essentially the rf SQUID described above. He schematically indicated how such devices could be inductively coupled to yield a shift register, with what he called a bifurcator, which is basically the same as the COPY gate described below and a majority (AND) gate. Later, Likharev also gave a conceptual description of a reversible computer based on an array of parametric quantrons slightly modified from his original suggestion [14]. In Figs. 1-2 we show COPY, NOT, and AND gates constructed from coupled rf SQUIDs, in the spirit of the parametric quantron structure described by Likharev. They are functionally equivalent to the three conceptual atomic gates that Lloyd [15] proposed as the basis for a general quantum computer. The qubits are circulating currents; the assignment of the |0〉 state is indicated on the diagrams. For example, the COPY gate in Fig.1 has two SQUID loops A and B that are externally biased with one-half of a flux quantum and inductively coupled together. The state of A may be prepared via the "In" coupling loop and a ‘copy’ pulse applied to the bias line of B causes B to become a copy of A. The quantum state of B can be brought into interaction with the qubit in another gate via the "Out" coupling loop or B can be reset to “0” by applying a ‘reset’ pulse (which is oppositely directed to a ‘copy’ pulse) via its flux bias input.. Note the opposite circulating current directions for the |0〉 states in A and B. An inverter, or NOT gate, may be built very much like the COPY gate but with the flux bias of SQUID B reversed. In this case B is put in the |1〉 state when A is in the |0〉 state
COPY: B = A A
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0
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In
Out
Fig. 1 The COPY gate employing coupled single junction SQUIDs.
NOT: B = A A
B
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0
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In
Out
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AND: B = A
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0
0
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C
B In
Out
In
Fig. 2 The NOT gate is a variation of the COPY gate. The AND gate is constructed of three coupled SQUIDs.
and vice versa. Coupling the gate output to other qubits and reset are the same as for the COPY gate. Finally, the AND gate has two inputs and one output. By appropriately biasing the three loops and adjusting the mutual inductances, the state of B may be arranged to be the logical AND of the states of A and C. Circuit simulations indicate that the gates demonstrate correct classical operation although many essential quantum mechanical questions are yet to be answered. For example, if SQUID A in the COPY gate were initially placed in a superposition of the |1〉 and |0〉 states, after B copies A would B indeed be the same superposition of |1〉 and |0〉 as was A initially, i.e., does the classical COPY gate also copy quantum superpositions without destroying them? Also, how is A affected by the copy operation? Is there a back action of B on A which destroys the coherence of the state of A? These are central questions in further investigations; however, if a gate does not display the correct classical behavior, it certainly will not have the desired quantum behavior. III. TECHNICAL ISSUES The feasibility of a superconducting implementation of quantum computation hinges on several challenging technical issues. The most obvious ones are the required operating temperature, which will certainly be in the milliKelvin range, and the necessary junction quality, i.e. the amount of leakage current , which is a measure of the degree of coupling of the qubit degree of freedom to single electron modes. In addition it is necessary to design the environment of the superconducting circuit to prevent external modes from coupling to the computation. This includes the current lines used for bias and control of the qubits, which must be very high impedance, even at high frequencies. Single-flux-quantum (SFQ) logic is an established experimental technology, in which magnetic flux are held and processed through a series of SQUID loops. In the following discussion of technical issues we will use SFQ logic as a convenient classical analogy to guide our discussion of the requirements for superconductor quantum computing. The quantum coherence of the qubits must be maintained long enough to complete the computation. The relevant figure of merit is the quantum dephasing time, tφ , divided by the gate switching time, ts . DiVincenzo [5] has collected or
evaluated such data for a variety of suggested qubits from electrons in solids to trapped ions. The ratios range from 103 up to 1013 for certain trapped ions. DiVincenzo used h/∆E, where ∆E is the energy splitting of the two level system, as an estimate of the time required to execute one quantum gate operation -- surely a very optimistic estimate. We can make a much firmer and more realistic estimate of the switching time of an rf SQUID. Today's laboratory RSFQ circuits have switching times of about 3 ps which is limited by the resistive shunts used to prevent junction hysteresis. To open the possibility of quantum coherent computation we must project the development of Josephson junctions small enough that resistive shunts are not required. Then the switching time is limited by the junction capacitance and the surrounding circuit, and should be designed to be about 1 ps. For guidance in estimating the phase coherence time we use the theoretical predictions for the parameters necessary to observe macroscopic quantum coherence [16]. We may write the predicted quantum dephasing time as, tφ = 2
h2
R 10mK R = 0.4msec 10 10 Ω T Φ 20 k B T
(1)
where R is the junction resistance. Extrapolating a few years into the future, we choose a temperature T approaching 5 milliKelvin and a resistance ratio of 1010 (this seems a very conservative estimate of future junction quality, judging from the discussion below) over the normal resistance of 20 Ω. Then the quantum dephasing time is on the order of 10 milliseconds and the ratio of tφ to the switching time ts is 1010. Error correction schemes that may reduce the required number of steps over which quantum coherence must be maintained to on the order of 105 have recently been invented [17] - [19]. The BCS theory predicts that the "leakage" current, the quasiparticle current below the gap voltage, tends exponentially to zero as the temperature is reduced. The ratio of critical current to sub-gap quasiparticle current is given by Ic = I sg
π ∆ ∆ kBT e 8k B T
.
(2)
For Nb-based Josephson junctions with ∆ = 1.5 mV this is 108 at 1 K and 1076 at 0.1 K. Real SIS Josephson junctions have much higher leakage current at low temperature than this BCS prediction. This is a function of fabrication technology. The quasiparticle current of the best Josephson junctions can be resolved into a BCS term and a fabrication-dependent leakage current from tunnel barrier defects [20]. Therefore, there appears to be no fundamental reason that the leakage current cannot be reduced to any desired level, with proper fabrication technology. Although the leakage current scales with the critical current of the junction, it is not possible to use junctions with extremely small critical currents because the SQUID loops must have LIc ~ Φ0. If Ic is small then the circuit inductors become undesirably large. Judging from classical RSFQ circuits, we choose Ic ~ 100 µA. Without resistive shunts, the junction capacitance and hence the area must be small. How small? In classical SFQ circuits the capacitance is chosen according to RC = LJ/R. In the absence of resistors, the equivalent function is performed by the external circuit -- actually the next Josephson junction device in the circuit -- which can be treated as a lossless
transmission line which carries away the currents. Using realistic parameters to estimate the characteristic impedance of this line gives a junction area of about (0.5 µm)2. This is smaller, but not too much smaller, than most of today's Josephson junctions, and it is conveniently large enough to prevent intrusive single electron charging effects. It follows that the critical current density should be about 40,000 A/cm2. This is larger, but not too much larger, than most of today's Josephson junctions. It is essential that the junction capacitance not be too small. For a BCS SIS junction the peak voltage is close to twice the energy gap voltage [21]. This voltage is high enough to produce quasiparticle currents and hence dissipation. It is necessary to chose the capacitance according to the conditions given above in order to reduce the peak voltage below the gap voltage, thus preventing dissipation. In other words, a reasonable junction design for a quantum computer would have a pulse height of 2 mV and hence a pulse width of 1 ps. The fabrication technology of Nb-based Josephson junctions has been improving at a remarkable rate in recent years. For instance, Josephson junctions with area 0.25 µm 2 and jc ~ 10,000 A/cm2 have been fabricated [22], and excellent results have been achieved with a junction with one dimension lithographically fabricated at 0.2 µm [23]. Also, junctions with a ratio of critical current to sub-gap quasiparticle current of 2 x 108 at 0.8 K are fabricated [20]. Other groups have similar capabilities (see for example [24] and [25]). Also, Josephson junctions with current density up to 400,000 A/cm2 have been fabricated [26]. Thus, each of the junction requirements for quantum computation can be met today. However, it may be much more difficult to meet all of these requirements in the same Josephson junction. Another consideration is magnetic coupling of the flux quanta to magnetic impurities in the physical circuit. The environment of magnetic impurities may serve to “make a measurement” on the qubit flux forcing it into an environment-selected state and thereby destroying the qubit superposition state. A recent report of the observation of the Aharonov-Casher effect in arrays of microfabricated Josephson junctions [27] indicates that quantum coherence of fluxons may be maintained in a superconducting integrated circuit of modest dimensions. IV. CONCLUSION If quantum coherent computation can ever be realized it may revolutionize the scope of computing by enabling computational tasks unimaginable with classical computers. Superconducting electronics appears to offer the essential elements of quantum computation, although this technology will need to be pushed to its extreme limits to realize the promise. In this paper it has been our goal simply to point out the potential for a superconducting electronic embodiment of quantum computation and to offer a preliminary assessment of the magnitude of the technical challenge to achieve it. Note, however, that many knowledgeable scientists believe that useful quantum computation is next to unattainable in any technology [28], [29]. REFERENCES [1] C. Monroe, D.M. Meekhof, B.E. King, W.M. Itano, and D.J. Wineland, “Demonstration of a fundamental quantum logic gate”, Physical Review Letters, vol.75, pp. 4714-17, Dec. 1995. [2] P. Domokos, J.M. Raimond, M. Brune, and S. Haroche, “Simple cavityQED two-bit universal quantum logic gate: the principle and expected performances”, Physical Review A, vol. 72, pp. 3554-9, Nov. 1995.
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