Quantum Computation – Future of Microelectronics?

P. Hawrylak

Institute for Microstructural Sciences
National Research Council of Canada, Ottawa, K1A OR6 Ontario, Canada

1.   Introduction

Current silicon technology has steadily improved our ability to compute by increasing the number of bits and gates. Yet with the existing technology many problems are likely to remain unsolved: properties of quantum materials, multiscale problems inherent to nanoscience, drug design and discovery, hard mathematical problems such as factorization of prime numbers essential for security, to name a few. Quantum instead of classical computation has been suggested as a possible solution.^1,2 We will attempt to present our perspective of how quantum computing might fit into microelectronics.

Quantum computation attempts to take advantage of the same property that makes some of the problems so difficult to solve – quantum-mechanical behavior of many-particle systems. In quantum mechanics the state of the system, e.g. composed of a number of electrons and nuclei, is described by a superposition of electronic configurations. Imagine such electronic configuration, |1,1,0,0,1,0,0,1,0,1,0,0, where N_e = 5 electrons are distributed on N_s = 12 possible states. Here the states could be atomic orbitals of the quantum material, but we can think of them equally well as “quantum registers”, where “0” means empty and “1” means occupied. The problem with simulating a quantum material is the number of possible configurations in which this material can be found. For N_s = 12 and N_e = 5, the number of configurations is 12!/[5!(12−5)!] = 792, a manageable number. However, doubling the number of both states and electrons to N_s = 24 and N_e = 10 gives rise to over a million of configurations. Increasing the number of states tenfold to 240, and the number of electrons to 100, leads to 10⁷⁰ possible configurations, i.e. a number comparable to the number of atoms on Earth. Hence it is clear that, on one hand, it is impossible to know all the quantum configurations of even a very small quantum system, and on the other hand, if we learn how to harness these quantum states new avenues for computation become possible. Here the abundance of configurations is treated as a resource and the computation is facilitated by interference and entanglement. Of course, these ideas are not new: they have long been studied in condensed matter physics as “correlated electron physics”. Here the entanglement and quantum properties of materials are due to “correlations” among electrons. Correlations found so far, such as superfluidity and superconductivity, exist only at very low temperatures. At higher temperatures, the correlations cease to be important and the classical behavior dominates. This brings us to a notion of decoherence. Decoherence implies interaction of the quantum system with environment and destruction of its quantum state. Hence, quantum computation, just like it classical counterpart, is prone to errors. Detecting errors requires measurement, and measurement destroys, or collapses, the quantum state. So, at a first glance, the error correction appears to be impossible in quantum computation. Answering this challenge, i.e. showing how to correct errors in a quantum computer, was a very important achievement.³ Error correction can be crudely described as correcting the phase of a qubit using auxiliary qubits, and correcting the amplitude by using classical error correcting means. While possible in principle, error correction leads to an enormous increase in complexity. Hence it seems we should initially attempt quantum information processing with devices that require only a small number of coherent qubits.

2.   Applications of few-qubit systems

There are several potential applications of few-qubit systems, from quantum economics, quantum cryptography, quantum metrology, to quantum nuclear magnetic resonance (NMR). Quantum economics is perhaps less known than the rest.⁴ It relies on quantum games and quantum strategies of conflict resolution.⁵ Quantum bidding is attractive because it involves only a few players and the payoff for the winner is high. An example of an experimentally realized two-qubit quantum game is the “prisoner dilemma”, where the prisoners chances of escape are enhanced by using a “quantum game” to reach cooperation.⁶

The quantum key distribution systems employed in quantum cryptography are another example of a few-qubit system. This field is very advanced, and a number of commercial quantum cryptography systems are already on the market. Here the main challenge is a technical one, i.e. generating reliable sources of single photons and pairs of entangled photons⁷ with wavelength compatible with current telecommunication wavelength of 1.5 μm.⁸

3.   Few-qubit systems based on electron spin

There are many proposals and several rudimentary implementations of a quantum computer. They include: (a) nuclear spin combined with commercial NMR techniques,⁹ (b) superconductive qubits,¹⁰ (c) atom and ion traps,¹¹ (e) polarization states of a photon and linear optics,¹² (f) topological qubits in fractional quantum Hall effect (FQHE),¹³ (g) solid state nuclear spins,¹⁴ and (h) quantum dot and electron-spin based qubits.^15,16 Here the references are not exhaustive but rather indicative of research activities in each of the subfields. Out of these different proposals, the quantum dot and electron-spin based implementation starts with the field effect transistor (FET) structures and appears the most compatible with current microelectronic technology. In this chapter we will explore whether microelectronics may evolve into quantum information technology, and investigate the current state-of-the-art. While the examples below will be drawn primarily from our own work, there is a significant and parallel effort worldwide.

With |0 and |1 states of the computational electron spin qubit basis equivalent to the two states of the electron spin S, the quantum computer Hamiltonian can be simply written as:

Here S_i is the spin of electron localized on the ith quantum dot, B_i is the local magnetic field acting on the ith spin and allowing single-qubit operations, and J_ij is the exchange coupling of two different spins. To build such a device one needs to develop technology that can: (a) localize electrons, (b) control precisely their numbers, (c) control their spin, and (d) control coupling between electrons. We like to think of this emerging enabling technology as nano-Spintronics.¹⁷ We now proceed to describe the current status of nano-spintronics with quantum dots.

   Single-electron spin qubit

A prerequisite for nano-spintronics is an ability to localize a single electron in a specific location, and “functionalize” this location. Localizing a single electron has been accomplished by using donors in semiconductors. However, the control of the position of individual donors and hence individual electrons is yet to be realized. Initial progress has been made with vertical quantum dots. Ashoori and co-workers build vertical quantum dots which were charged with few electrons.¹⁸ Their properties were measured and understood using single electron capacitance spectroscopy.^18,19 In the next step, Tarucha and co-workers demonstrated a vertical quantum dot device with electron numbers down to one.²⁰ This device was connected both to source and drain, and the electronic states of this device were measured using the Coulomb blockade spectroscopy. The scalability of single electron devices is, however, best accomplished by using lateral metallic gates in a planar technology familiar in the microelectronics. Prior to 1999, lateral gating was not entirely successful in controlling electron numbers, but in 1999 Hawrylak and co-workers showed how to use spin flips to determine the number of electrons that are already in the lateral quantum dot.²¹ This theory and experiment suggested that the application of moderate magnetic fields leads to spin polarization of edge states in both source and drain. Hence, a lateral quantum dot can be connected to spin-polarized contacts and current through such a device can be blocked not only by the charge (as in Coulomb blockade) but also by the spin – hence the concept of spin blockade spectroscopy (SBS). In the following year, Sachrajda and coworkers emptied a lateral quantum dot,²² shown in Fig. 1, and filled it with a controlled number of electrons. The properties of such a device were probed by SBS and it was demonstrated that one can manipulate the spin state of a quantum dot. When a dot was connected to spin-polarized leads, the current was switched on and off by changing the spin state of the quantum dot device at a single-electron level, demonstrating the single spin transistor.²³ The lateral quantum dot device shown in Fig.1 allows to localize a single electron spin and represents a single qubit.

Figure 1. Schematic structure (a) and the energy profile (b) of the single-electron lateral quantum dot with controlled electron numbers.

   Two-electron spin qubits

In Fig. 2(a) we show a double-dot device,^24-27 studied by Pioro-Ladriere et al.,^24,25 which makes it possible to localize two electrons in two specific locations, as indicated by arrows. A quantum point contact (QPC) has been build in the vicinity of the double dot device. The current I_QPC through the QPC is a sensitive function of total charge and its distribution inside the double quantum dot: the closer the electrons in a double dot are to the QPC, the smaller the current. Figure 2(b) shows the stability diagram of the double dot device,²⁵ i.e. the derivative of QPC current as a function of two plunger gate voltages, after Ref. 25. Changing the potential applied to the gates changes the electron numbers and their location in the device. In Fig. 2(b), lines correspond to changes in total electron numbers, and different slope of these lines allows us to deduce in which quantum dot the electrons can be found. In this way. the quantum-dot occupation numbers N₁ and N₂ can be assigned to dot 1 and dot 2. We see that one can have no electrons in the device (0,0), one electron in the left dot (1,0) or one electron in the right dot (0,1), and one electron in each dot (1,1).

The (1,1) configuration is schematically represented by arrows in Fig. 2(a). One of the objectives is the ability to hybridize the two dots, i.e. to control the exchange coupling constant J_ij in Eq. (1). There are two measures of hybridization: the quantum-mechanical tunneling t of a single electron between the two dots, and the exchange coupling J_ij for two electrons.

Figure 2. Lateral quantum dot implementation of the two-qubit system. (a) SEM picture of metallic gates of a double dot device integrated with a quantum point contact (QPC) charge readout device. (b) Charging diagram of a double dot device – derivative of the I_QPC as a function of plunger gate voltages for a fixed voltage applied to the barrier gate; (N₁, N₂) indicate electron numbers in dot 1 and dot 2.

The single-particle tunneling contributes to the exchange coupling via the super-exchange ~2t²/U, where U is the on-site Coulomb energy. The quantum-mechanical tunneling is indirectly responsible for the width of the transition region between the (1,0) and (0,1) configurations: the stronger the tunneling, the wider and smoother the transition. Typical values of t measured in lateral dots approach 100 μeV. A number of groups have measured and tuned the exchange coupling J by applying bias to one of the dots. Moreover, Petta et al. demonstrated coherent Rabi oscillations between the singlet and one of the triplet two-electron states.²⁸ The two states served as the two states of a “coded qubit”. However, the operation of a double dot as a controlled-NOT (CNOT) gate remains to be demonstrated.

   Three-electron spin qubits

The double quantum dot device is the first step toward a scalable quantum processor. It can, however, serve at best as a CNOT gate. The realization of a class of nontrivial algorithms, such as quantum teleportation,²⁹ requires at least three spins.³⁰ It has been also shown that three spins are needed to realize a coded qubit,^31,32 which combines long coherence times associated with electron spin with ease of operation associated with voltages. Inset to Fig. 3(a) shows both a schematic representation of a triple-dot device with one spin per dot, as well as the SEM picture of a triple dot device investigated by Gaudreau et al.³³ This triple-dot device combines a double-dot metallic gate layout with an impurity located in the left dot. The presence of three potential minima being filled with electrons localized in three spatially distinct regions is demonstrated by three different slopes of addition lines shown in Fig. 3(a). The three families of lines correspond to electron addition to a particular dot. An analysis similar to our treatment of the double-dot device allows us to assign particle numbers (N₁,N₂,N₃) for each of the dots. One can empty the device, i.e. drive it into the (0,0,0) state, and then fill it up with three electrons, one electron per dot, indicated by the (1,1,1) region in Fig. 3(a). Figure 3(b) shows the manipulation of the three-electron complex using gate voltages V_A and V_B. The goal here is to bring the three dots into resonance and manipulate their exchange coupling. This is difficult because it is hard to visualize, understand, and control a complex network with just two external gate voltages.

While we are focusing here on quantum information, the fabrication of artificial networks, where electrons are localized in specific locations, and tunnel from dot to dot, is equivalent to the experimental realization of the Hubbard model. The physics of the Hubbard model is a cornerstone of the physics of “quantum materials”,^34–36 from high-T_C superconductors to unusual magnetic properties of oxides. We feel that the artificial quantum materials on a chip, which one is beginning to explore, will be very useful in answering many of the outstanding questions in the physics of quantum materials.

Figure 3. Triple-dot implementation of the three-qubit system: (a) stability diagram showing two dots on resonance, with inset showing an SEM picture of device, including gates A and B together with the I_QPC; (b) stability diagram and charge reorganization induced by gate voltages of a three-electron complex.³³

4.   Challenges in electron-spin qubits and conclusions

There has been a significant progress in the realization of electron-spin based qubits, as summarized above. A single-, double-, and a three-qubit systems have been realized in lateral quantum dots, starting from a FET structure. Single qubit operation has been demonstrated very recently by Koppens et al.³⁷ This progress bodes well for future few-qubit systems compatible with microelectronics.

However, there are still many challenges left. In particular, the CNOT gate still needs to be demonstrated. The CNOT gate requires both the ability to control the exchange coupling, as well as the single qubit operation. The CNOT gate, when integrated into the triple dot design, would allow the realization of simple algorithms. In addition, the problem of decoherence, and in particular the coupling of electron and nuclear spins, needs addressing. Perhaps a solution might lie in carbon-based quantum dot systems. Recent incorporation of graphene into FET structures opens up this possibility.³⁸

Finally, the progress in single photon sources and sources of entangled photon pair production looks promising for quantum cryptography.

Acknowledgments

This work was supported in part by the Canadian Institute for Advanced Research. I thank A. Sachrajda, M. Korkusinski, and R. Williams for collaboration.

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