Alexander N. Korotkov

Superconducting quantum circuits at the surface code threshold for fault tolerance

A quantum computer can solve hard problems, such as prime factoring, database searching and quantum simulation, at the cost of needing to protect fragile quantum states from error. Quantum error correction provides this protection by distributing a logical state among many physical quantum bits (qubits) by means of quantum entanglement. Superconductivity is a useful phenomenon in this regard, because it allows the construction of large quantum circuits and is compatible with microfabrication. For superconducting qubits, the surface code approach to quantum computing is a natural choice for error correction, because it uses only nearest-neighbour coupling and rapidly cycled entangling gates. The gate fidelity requirements are modest: the per-step fidelity threshold is only about 99 per cent. Here we demonstrate a universal set of logic gates in a superconducting multi-qubit processor, achieving an average single-qubit gate fidelity of 99.92 per cent and a two-qubit gate fidelity of up to 99.4 per cent. This places Josephson quantum computing at the fault-tolerance threshold for surface code error correction. Our quantum processor is a first step towards the surface code, using five qubits arranged in a linear array with nearest-neighbour coupling. As a further demonstration, we construct a five-qubit Greenberger–Horne–Zeilinger state using the complete circuit and full set of gates. The results demonstrate that Josephson quantum computing is a high-fidelity technology, with a clear path to scaling up to large-scale, fault-tolerant quantum circuits.A quantum computer can solve hard problems - such as prime factoring, database searching, and quantum simulation - at the cost of needing to protect fragile quantum states from error. Quantum error correction provides this protection, by distributing a logical state among many physical qubits via quantum entanglement. Superconductivity is an appealing platform, as it allows for constructing large quantum circuits, and is compatible with microfabrication. For superconducting qubits the surface code is a natural choice for error correction, as it uses only nearest-neighbour coupling and rapidly-cycled entangling gates. The gate fidelity requirements are modest: The per-step fidelity threshold is only about 99%. Here, we demonstrate a universal set of logic gates in a superconducting multi-qubit processor, achieving an average single-qubit gate fidelity of 99.92% and a two-qubit gate fidelity up to 99.4%. This places Josephson quantum computing at the fault-tolerant threshold for surface code error correction. Our quantum processor is a first step towards the surface code, using five qubits arranged in a linear array with nearest-neighbour coupling. As a further demonstration, we construct a five-qubit Greenberger-Horne-Zeilinger (GHZ) state using the complete circuit and full set of gates. The results demonstrate that Josephson quantum computing is a high-fidelity technology, with a clear path to scaling up to large-scale, fault-tolerant quantum circuits.

Logic gates at the surface code threshold: Superconducting qubits poised for fault-tolerant quantum computing

A quantum computer can solve hard problems, such as prime factoring, database searching and quantum simulation, at the cost of needing to protect fragile quantum states from error. Quantum error correction provides this protection by distributing a logical state among many physical quantum bits (qubits) by means of quantum entanglement. Superconductivity is a useful phenomenon in this regard, because it allows the construction of large quantum circuits and is compatible with microfabrication. For superconducting qubits, the surface code approach to quantum computing is a natural choice for error correction, because it uses only nearest-neighbour coupling and rapidly cycled entangling gates. The gate fidelity requirements are modest: the per-step fidelity threshold is only about 99 per cent. Here we demonstrate a universal set of logic gates in a superconducting multi-qubit processor, achieving an average single-qubit gate fidelity of 99.92 per cent and a two-qubit gate fidelity of up to 99.4 per cent. This places Josephson quantum computing at the fault-tolerance threshold for surface code error correction. Our quantum processor is a first step towards the surface code, using five qubits arranged in a linear array with nearest-neighbour coupling. As a further demonstration, we construct a five-qubit Greenberger–Horne–Zeilinger state using the complete circuit and full set of gates. The results demonstrate that Josephson quantum computing is a high-fidelity technology, with a clear path to scaling up to large-scale, fault-tolerant quantum circuits.A quantum computer can solve hard problems - such as prime factoring, database searching, and quantum simulation - at the cost of needing to protect fragile quantum states from error. Quantum error correction provides this protection, by distributing a logical state among many physical qubits via quantum entanglement. Superconductivity is an appealing platform, as it allows for constructing large quantum circuits, and is compatible with microfabrication. For superconducting qubits the surface code is a natural choice for error correction, as it uses only nearest-neighbour coupling and rapidly-cycled entangling gates. The gate fidelity requirements are modest: The per-step fidelity threshold is only about 99%. Here, we demonstrate a universal set of logic gates in a superconducting multi-qubit processor, achieving an average single-qubit gate fidelity of 99.92% and a two-qubit gate fidelity up to 99.4%. This places Josephson quantum computing at the fault-tolerant threshold for surface code error correction. Our quantum processor is a first step towards the surface code, using five qubits arranged in a linear array with nearest-neighbour coupling. As a further demonstration, we construct a five-qubit Greenberger-Horne-Zeilinger (GHZ) state using the complete circuit and full set of gates. The results demonstrate that Josephson quantum computing is a high-fidelity technology, with a clear path to scaling up to large-scale, fault-tolerant quantum circuits.

Single‐electron transistor logic

We present the results of numerical simulations of a functionally complete set of complementary logic circuits based on capacitively coupled single‐electron transistors (CSETs). The family includes an inverter/buffer stage, as well as two‐input NOR, NAND, and XOR gates, all using similar tunnel junctions, and the same dc bias voltage and logic levels. Maximum operation temperature, switching speed, power consumption, noise tolerances, error rate, and critical parameter margins of the basic gates have been estimated. When combined with the data from a preliminary geometrical analysis, the results indicate that implementation of the CSET logic family for operation at T∼20 K will require fabrication of structures with ∼2‐nm‐wide islands separated by ∼1‐nm‐wide tunnel gaps.

Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback

The act of measurement bridges the quantum and classical worlds by projecting a superposition of possible states into a single, albeit probabilistic, outcome. The time-scale of this “instantaneous” process can be stretched using weak measurements so that it takes the form of a gradual random walk towards a final state. Remarkably, the interim measurement record is sufficient to continuously track and steer the quantum state using feedback. Here, we report the first implementation of quantum feedback control in a solid state system, in our case a superconducting quantum bit (qubit) coupled to a microwave cavity. Probing the state of the cavity with less than one photon on average, implements a weak measurement of the qubit state. These photons are then directed to a high-bandwidth quantum-noise-limited amplifier, which enables realtime monitoring of the state of the cavity—and hence that of the qubit—with high fidelity. We demonstrate quantum feedback control by inhibiting the decay of Rabi oscillations, allowing them to persist indefinitely. This new ability permits active suppression of decoherence and defines a path for continuous quantum error correction. Other novel avenues include quantum state stabilization, entanglement generation using measurement, state purification, and adaptive measurements.The act of measurement bridges the quantum and classical worlds by projecting a superposition of possible states into a single (probabilistic) outcome. The timescale of this ‘instantaneous’ process can be stretched using weak measurements, such that it takes the form of a gradual random walk towards a final state. Remarkably, the interim measurement record is sufficient to continuously track and steer the quantum state using feedback. Here we implement quantum feedback control in a solid-state system, namely a superconducting quantum bit (qubit) coupled to a microwave cavity. A weak measurement of the qubit is implemented by probing the cavity with microwave photons, maintaining its average occupation at less than one photon. These photons are then directed to a high-bandwidth, quantum-noise-limited amplifier, which allows real-time monitoring of the state of the cavity (and, hence, that of the qubit) with high fidelity. We demonstrate quantum feedback control by inhibiting the decay of Rabi oscillations, allowing them to persist indefinitely. Such an ability permits the active suppression of decoherence and enables a method of quantum error correction based on weak continuous measurements. Other applications include quantum state stabilization, entanglement generation using measurement, state purification and adaptive measurements.

Implementing the Quantum von Neumann Architecture with Superconducting Circuits

A quantum version of a central processing unit was created with superconducting circuits and elements. The von Neumann architecture for a classical computer comprises a central processing unit and a memory holding instructions and data. We demonstrate a quantum central processing unit that exchanges data with a quantum random-access memory integrated on a chip, with instructions stored on a classical computer. We test our quantum machine by executing codes that involve seven quantum elements: Two superconducting qubits coupled through a quantum bus, two quantum memories, and two zeroing registers. Two vital algorithms for quantum computing are demonstrated, the quantum Fourier transform, with 66% process fidelity, and the three-qubit Toffoli-class OR phase gate, with 98% phase fidelity. Our results, in combination especially with longer qubit coherence, illustrate a potentially viable approach to factoring numbers and implementing simple quantum error correction codes.

CONTINUOUS QUANTUM MEASUREMENT OF A DOUBLE DOT

We consider the continuous measurement of a double quantum dot by a weakly coupled detector ~tunnel point contact nearby!. While the conventional approach describes the gradual system decoherence due to the measurement, we study the situation when the detector output is explicitly recorded that leads to the opposite effect: gradual purification of the double-dot density matrix. The nonlinear Langevin equation is derived for the random evolution of the density matrix which is reflected and caused by the stochastic detector output. Gradual collapse, gradual purification, and the quantum Zeno effect are naturally described by the equation. We also discuss the possible experiments to confirm the theory. @S0163-1829~99!01032-2# The problem of quantum measurements has a long history; however, it still attracts considerable attention and even causes some controversy, mainly concerning the wavefunction ‘‘collapse’’ ~see, e.g., Refs. 1 and 2!. Among various modern approaches to this problem let us mention the idea of replacing the collapse postulate by the gradual decoherence of the density matrix due to the interaction with the detector 3 and the approach of a stochastic evolution of the wave function ~see, e.g., Refs. 4‐11!. The latter approach ~which is used in the present paper! can describe the selective measurements for which the system evolution is conditioned on the particular measurement result ~other keywords of the approach are quantum trajectories, quantum state diffusion, quantum jumps, etc.!. The renewed interest in the measurement problem is justified by the development of experimental technique, which allows more and more experimental studies of quantum measurement in optics and mesoscopic structures. 12‐17 The problem also has a close connection to the rapidly growing fields of quantum cryptography and quantum computing. 18 In the recent experiment 15 with the ‘‘which-path’’ interferometer the suppression of Aharonov-Bohm interference due to the detection of which path an electron chooses, was observed. The weakly coupled quantum point contact was used as a detector. The interference suppression in this experiment can be quantitatively described by the decoherence due to the measurement process. 19‐22 We will consider a somewhat different setup: two quantum dots occupied by one electron and a weakly coupled detector ~point contact nearby! measuring the position of the electron. The decoherence of the double-dot density matrix due to continuous measurement in this setup has been analyzed in Refs. 19 and 22. However, the decoherence approach cannot describe the detector output which is a separate problem analyzed in the present paper. We answer two interrelated questions: how the detector current behaves in time and what is the proper double-dot density matrix for a particular detector output. We show that the models of point contact considered in Refs. 19‐21 describe an ideal detector. In this case the density matrix decoherence is just a consequence of averaging over all possible measurement results. For any particular detector output our equations allow the evolution of pure wave function to be followed. Moreover, a mixed density matrix can be gradually purified in the course of a continuous measurement.

Experimental violation of a Bell’s inequality in time with weak measurement

Macrorealism assumes that a macroscopic object is at any given time in one of the distinct states it has available, and that it is in principle possible to determine which state the system is in without disturbing its dynamics. An experiment now demonstrates that a superconducting microelectronic system violates macrorealism and obeys the laws of quantum mechanics.

Possible performance of capacitively coupled single‐electron transistors in digital circuits

We have carried out a theoretical analysis of the possible performance of single‐electron transistors with capacitive coupling in simple logic and memory circuits. Both resistively loaded and complementary transistors have been analyzed, with a detailed account of parasitic factors including thermal fluctuations and background charge variations. The analysis shows that at optimal values of the parameters including the background charge, the maximum operation temperature is close to 0.025e2/CkB, where C is the capacitance of the smallest tunnel junction. At T∼0.01e2/CkB the parameter margins are relatively wide; for the structures with 2‐nm minimum feature size, the latter temperature is close to 77 K. A typical margin for background charge fluctuations is on the order of 0.1e; these fluctuations may be a major obstacle for practical ultradense single‐electron circuits.

A NUMERICAL STUDY OF THE DYNAMICS AND STATISTICS OF SINGLE ELECTRON SYSTEMS

We describe a new and efficient method for the numerical study of the dynamics and statistics of single electron systems presenting arbitrary combinations of small tunnel junctions, capacitances, and voltage sources. The method is based on numerical solution of a linear matrix equation for the vector of probabilities of various electric charge states of the system, with iterative refining of the operational set of states. The method is able to describe very small deviations from the ‘‘classical’’ behavior of a system, due to the finite speed of applied signals, thermal activation, and macroscopic quantum tunneling of charge (cotunneling). As an illustration, probability of dynamic and static errors in two single electron memory cells with 6 and 8 tunnel junctions have been calculated as a function of bias voltage, temperature, and switching speed.

Decoherence suppression by quantum measurement reversal

We show that the qubit decoherence due to zero-temperature energy relaxation can be almost completely suppressed by using the quantum uncollapsing procedure. To protect a qubit state, a partial quantum measurement moves it towards the ground state, where it is kept during the storage period, while the second partial measurement restores the initial state. This procedure preferentially selects the cases without energy decay events. Stronger decoherence suppression requires smaller selection probability; a desired point in this trade-off can be chosen by varying the measurement strength. The experiment can be realized in a straightforward way using the superconducting phase qubit.