Konstantin Nesterov

We describe the generation of entangling gates on superconductor-semiconductor hybrid qubits by ac voltage modulation of the Josephson energy. Our numerical simulations demonstrate that the unitary error can be below 10^{−5} in a variety of 75-ns-long two-qubit gates (CZ, iSWAP, and \sqrt{iSWAP}) implemented using parametric resonance. We analyze the conditional ZZ phase and demonstrate that the CZ gate needs no further phase correction steps, while the ZZ phase error in SWAP-type gates can be… Show more

We describe the generation of entangling gates on superconductor-semiconductor hybrid qubits by ac voltage modulation of the Josephson energy. Our numerical simulations demonstrate that the unitary error can be below 10^{−5} in a variety of 75-ns-long two-qubit gates (CZ, iSWAP, and \sqrt{iSWAP}) implemented using parametric resonance. We analyze the conditional ZZ phase and demonstrate that the CZ gate needs no further phase correction steps, while the ZZ phase error in SWAP-type gates can be compensated by choosing pulse parameters. With decoherence considered, we estimate that qubit relaxation time needs to exceed 70μs to achieve the 99.9% fidelity threshold. Show less

The superconducting fluxonium qubit has a great potential for high-fidelity quantum gates with its long coherence times and strong anharmonicity at the half flux quantum sweet spot. However, current implementations of two-qubit gates compromise fluxonium's coherence properties by requiring either a temporary population of the non-computational states or tuning the magnetic flux off the sweet spot. Here we realize a fast all-microwave cross-resonance gate between two capacitively-coupled… Show more

The superconducting fluxonium qubit has a great potential for high-fidelity quantum gates with its long coherence times and strong anharmonicity at the half flux quantum sweet spot. However, current implementations of two-qubit gates compromise fluxonium's coherence properties by requiring either a temporary population of the non-computational states or tuning the magnetic flux off the sweet spot. Here we realize a fast all-microwave cross-resonance gate between two capacitively-coupled fluxoniums with the qubit dynamics well confined to the computational space. We demonstrate a direct CNOT gate in 70 ns with fidelity up to 0.9949(6) despite the limitations of a sub-optimal measurement setup and device coherence. Our results project a possible pathway towards reducing the two-qubit error rate below 0.0001 with present-day technologies. Show less

In two-dimensional electron systems with broken inversion and time-reversal symmetries, a Josephson junction reveals an anomalous response: the supercurrent is nonzero even at zero phase difference between two superconductors. We consider details of this peculiar phenomenon in the planar double-barrier configurations of hybrid circuits, where the noncentrosymmetric normal region is described in terms of the paradigmatic Rashba model of spin-orbit coupling. We analyze this anomalous Josephson… Show more

In two-dimensional electron systems with broken inversion and time-reversal symmetries, a Josephson junction reveals an anomalous response: the supercurrent is nonzero even at zero phase difference between two superconductors. We consider details of this peculiar phenomenon in the planar double-barrier configurations of hybrid circuits, where the noncentrosymmetric normal region is described in terms of the paradigmatic Rashba model of spin-orbit coupling. We analyze this anomalous Josephson effect by means of both the Ginzburg-Landau formalism and the microscopic Green's functions approach in the clean limit. The magnitude of the critical current is calculated for an arbitrary in-plane magnetic field orientation, and anomalous phase shifts in the Josephson current-phase relation are determined in terms of the parameters of the model in several limiting cases. Show less

We analyze the cross-resonance effect for fluxonium circuits and investigate a two-qubit gate scheme based on selective darkening of a transition. In this approach, two microwave pulses at the frequency of the target qubit are applied simultaneously with a proper ratio between their amplitudes to achieve a controlled-not operation. We study in detail coherent gate dynamics and calculate gate error. With nonunitary effects accounted for, we demonstrate that gate error below 0.0001 is possible… Show more

We analyze the cross-resonance effect for fluxonium circuits and investigate a two-qubit gate scheme based on selective darkening of a transition. In this approach, two microwave pulses at the frequency of the target qubit are applied simultaneously with a proper ratio between their amplitudes to achieve a controlled-not operation. We study in detail coherent gate dynamics and calculate gate error. With nonunitary effects accounted for, we demonstrate that gate error below 0.0001 is possible for realistic hardware parameters. This number is facilitated by long coherence times of computational transitions and strong anharmonicity of fluxoniums, which easily prevents excitation to higher excited states during the gate microwave drive. Show less

We analyze a high-fidelity two-qubit gate using fast flux pulses on superconducting fluxonium qubits. The gate is realized by temporarily detuning the magnetic flux through the fluxonium loop away from the half flux quantum sweet spot. We simulate dynamics of two capacitively coupled fluxoniums during the flux pulses and optimize the pulse parameters to obtain a highly accurate sqrt(iswap)-like entangling gate. We also evaluate the effect of the flux noise and qubit relaxation on the gate… Show more

We analyze a high-fidelity two-qubit gate using fast flux pulses on superconducting fluxonium qubits. The gate is realized by temporarily detuning the magnetic flux through the fluxonium loop away from the half flux quantum sweet spot. We simulate dynamics of two capacitively coupled fluxoniums during the flux pulses and optimize the pulse parameters to obtain a highly accurate sqrt(iswap)-like entangling gate. We also evaluate the effect of the flux noise and qubit relaxation on the gate fidelity. Our results demonstrate that the gate error remains below 0.0001 for a currently achievable magnitude of the flux noise and qubit relaxation time. Show less

Transforming stand-alone qubits into a functional, general-purpose quantum processing unit requires an architecture where many-body quantum entanglement can be generated and controlled in a coherent, modular, and measurable fashion. Electronic circuits promise a well-developed pathway for large-scale integration once a mature library of quantum-compatible elements have been developed. In the domain of superconducting circuits, fluxonium has recently emerged as a promising qubit due to its… Show more

Transforming stand-alone qubits into a functional, general-purpose quantum processing unit requires an architecture where many-body quantum entanglement can be generated and controlled in a coherent, modular, and measurable fashion. Electronic circuits promise a well-developed pathway for large-scale integration once a mature library of quantum-compatible elements have been developed. In the domain of superconducting circuits, fluxonium has recently emerged as a promising qubit due to its high-coherence and large anharmonicity, yet its scalability has not been systematically explored. In this work, we present a blueprint for a high-performance fluxonium-based quantum processor that addresses the challenges of frequency crowding, and both quantum and classical crosstalk. The main ingredients of this architecture include high-anharmonicity circuits, multipath couplers to entangle qubits where spurious longitudinal coupling can be nulled, circuit designs that are compatible with multiplexed microwave circuitry, and strongly coupled readout channels that do not require complex, frequency-sculpted elements to maintain coherence. In addition, we explore robust and resource-efficient protocols for quantum logical operations, then perform numerical simulations to validate the expected performance of this proposed processor with respect to gate fidelity, fabrication yield, and logical error suppression. Lastly, we discuss practical considerations to implement the architecture and achieve the anticipated performance. Show less

Large scale quantum computing motivates the invention of two-qubit gate schemes that not only maximize the gate fidelity but also draw minimal resources. In the case of superconducting qubits, the weak anharmonicity of transmons imposes profound constraints on the gate design, leading to increased complexity of devices and control protocols. Here we demonstrate a resource-efficient control over the interaction of strongly-anharmonic fluxonium qubits. Namely, applying an off-resonant drive to… Show more

Large scale quantum computing motivates the invention of two-qubit gate schemes that not only maximize the gate fidelity but also draw minimal resources. In the case of superconducting qubits, the weak anharmonicity of transmons imposes profound constraints on the gate design, leading to increased complexity of devices and control protocols. Here we demonstrate a resource-efficient control over the interaction of strongly-anharmonic fluxonium qubits. Namely, applying an off-resonant drive to noncomputational transitions in a pair of capacitively-coupled fluxoniums induces a ZZ interaction due to unequal ac Stark shifts of the computational levels. With a continuous choice of frequency and amplitude, the drive can either cancel the static ZZ term or increase it by an order of magnitude to enable a controlled-phase (CP) gate with an arbitrary programmed phase shift. The cross-entropy benchmarking of these non-Clifford operations yields a sub 1% error, limited solely by incoherent processes. Our result demonstrates the advantages of strongly-anharmonic circuits over transmons in designing the next generation of quantum processors. Show less

We propose a family of microwave-activated entangling gates on two capacitively coupled fluxonium qubits. A microwave pulse applied to either qubit at a frequency near the half-frequency of the |00⟩−|11⟩ transition induces two-photon Rabi oscillations with a negligible leakage outside the computational subspace, owing to the strong anharmonicity of fluxoniums. By adjusting the drive frequency, amplitude, and duration, we obtain the gate family that is locally equivalent to the… Show more

We propose a family of microwave-activated entangling gates on two capacitively coupled fluxonium qubits. A microwave pulse applied to either qubit at a frequency near the half-frequency of the |00⟩−|11⟩ transition induces two-photon Rabi oscillations with a negligible leakage outside the computational subspace, owing to the strong anharmonicity of fluxoniums. By adjusting the drive frequency, amplitude, and duration, we obtain the gate family that is locally equivalent to the fermionic-simulation gates such as SWAP‾‾‾‾‾‾‾√-like and controlled-phase gates. The gate error can be tuned below 10−4 for a pulse duration under 100 ns without excessive circuit parameter matching. Given that the fluxonium coherence time can exceed 1 ms, our gate scheme is promising for large-scale quantum processors. Show less

We demonstrate a controlled-Z gate between capacitively coupled fluxonium qubits with transition frequencies 72.3 MHz and 136.3 MHz. The gate is activated by a 61.6 ns long pulse at the frequency between non-computational transitions |10⟩−|20⟩ and |11⟩−|21⟩, during which the qubits complete only 4 and 8 Larmor periods, respectively. The measured gate error of (8±1)×10−3 is limited by decoherence in the non-computational subspace, which will likely improve in the next generation devices… Show more

We demonstrate a controlled-Z gate between capacitively coupled fluxonium qubits with transition frequencies 72.3 MHz and 136.3 MHz. The gate is activated by a 61.6 ns long pulse at the frequency between non-computational transitions |10⟩−|20⟩ and |11⟩−|21⟩, during which the qubits complete only 4 and 8 Larmor periods, respectively. The measured gate error of (8±1)×10−3 is limited by decoherence in the non-computational subspace, which will likely improve in the next generation devices. Although our qubits are about fifty times slower than transmons, the two-qubit gate is faster than microwave-activated gates on transmons, and the gate error is on par with the lowest reported. Architectural advantages of low-frequency fluxoniums include long qubit coherence time, weak hybridization in the computational subspace, suppressed residual ZZ-coupling rate (here 46 kHz), and absence of either excessive parameter matching or complex pulse shaping requirements. Show less

In superconducting circuit architectures for quantum computing, microwave resonators are often used both to isolate qubits from the electromagnetic environment and to facilitate qubit state readout. We analyze the full counting statistics of photons emitted from such driven readout resonators both in and beyond the dispersive approximation. We calculate the overlap between emitted-photon distributions for the two qubit states and explore strategies for its minimization with the purpose of… Show more

In superconducting circuit architectures for quantum computing, microwave resonators are often used both to isolate qubits from the electromagnetic environment and to facilitate qubit state readout. We analyze the full counting statistics of photons emitted from such driven readout resonators both in and beyond the dispersive approximation. We calculate the overlap between emitted-photon distributions for the two qubit states and explore strategies for its minimization with the purpose of increasing fidelity of intensity-sensitive readout techniques. In the dispersive approximation and at negligible qubit relaxation, both distributions are Poissonian, and the overlap between them can be easily made arbitrarily small. Nondispersive terms of the Hamiltonian generate squeezing and the Purcell decay with the latter effect giving the dominant contribution to the overlap between two distributions. Show less

The superconducting fluxonium circuit is an artificial atom with a strongly anharmonic spectrum: When biased at a half flux quantum, the lowest qubit transition is an order of magnitude smaller in frequency than those to higher levels. Similar to conventional atomic systems, such a frequency separation between the computational and noncomputational subspaces allows independent optimizations of the qubit coherence and two-qubit interactions. Here, we describe a controlled-
Z
gate for two… Show more

The superconducting fluxonium circuit is an artificial atom with a strongly anharmonic spectrum: When biased at a half flux quantum, the lowest qubit transition is an order of magnitude smaller in frequency than those to higher levels. Similar to conventional atomic systems, such a frequency separation between the computational and noncomputational subspaces allows independent optimizations of the qubit coherence and two-qubit interactions. Here, we describe a controlled-
Z
gate for two fluxoniums connected either capacitively or inductively, with qubit transitions fixed near
500
MHz
. The gate is activated by a microwave drive at a resonance involving the second excited state. We estimate intrinsic gate fidelities over 99.9% with gate times below 100 ns. Show less

Fast, high-fidelity measurement is a key ingredient for quantum error correction. Conventional approaches to the measurement of superconducting qubits, involving linear amplification of a microwave probe tone followed by heterodyne detection at room temperature, do not scale well to large system sizes. We introduce an approach to measurement based on a microwave photon counter demonstrating raw single-shot measurement fidelity of 92%. Moreover, the intrinsic damping of the photon counter is… Show more

Fast, high-fidelity measurement is a key ingredient for quantum error correction. Conventional approaches to the measurement of superconducting qubits, involving linear amplification of a microwave probe tone followed by heterodyne detection at room temperature, do not scale well to large system sizes. We introduce an approach to measurement based on a microwave photon counter demonstrating raw single-shot measurement fidelity of 92%. Moreover, the intrinsic damping of the photon counter is used to extract the energy released by the measurement process, allowing repeated high-fidelity quantum nondemolition measurements. Our scheme provides access to the classical outcome of projective quantum measurement at the millikelvin stage and could form the basis for a scalable quantum-to-classical interface. Show less

We review interaction effects in chaotic metallic nanoparticles.
Their single-particle Hamiltonian is described by the proper
random-matrix ensemble while the dominant interaction
terms are invariants under a change of the single-particle basis.
In the absence of spin-orbit scattering, the nontrivial invariants
consist of a pairing interaction, which leads to superconduc-
tivity in the bulk, and a ferromagnetic exchange interaction.
Spin-orbit scattering breaks… Show more

We review interaction effects in chaotic metallic nanoparticles.
Their single-particle Hamiltonian is described by the proper
random-matrix ensemble while the dominant interaction
terms are invariants under a change of the single-particle basis.
In the absence of spin-orbit scattering, the nontrivial invariants
consist of a pairing interaction, which leads to superconduc-
tivity in the bulk, and a ferromagnetic exchange interaction.
Spin-orbit scattering breaks spin-rotation invariance and when
it is sufficiently strong, the only dominant nontrivial interac-
tion is the pairing interaction. We discuss how the magnetic re-
sponse of discrete energy levels of the nanoparticle (which can
be measured in single-electron tunneling spectroscopy exper-
iments) is affected by such pairing correlations and how it can
provide a signature of pairing correlations. We also consider the
spin susceptibility of the nanoparticle and discuss how spin-
orbit scattering changes the signatures of pairing correlations
in this observable. Show less

We study Josephson junctions made of semiconducting nanowires with Rashba spin-orbit coupling, where superconducting correlations are induced by the proximity effect. In the presence of a suitably directed magnetic field, the system displays the anomalous Josephson effect: a nonzero supercurrent in the absence of a phase bias between two superconductors. We show that this anomalous current can be increased significantly by tuning the nanowire into the helical regime. In particular, in a short… Show more

We study Josephson junctions made of semiconducting nanowires with Rashba spin-orbit coupling, where superconducting correlations are induced by the proximity effect. In the presence of a suitably directed magnetic field, the system displays the anomalous Josephson effect: a nonzero supercurrent in the absence of a phase bias between two superconductors. We show that this anomalous current can be increased significantly by tuning the nanowire into the helical regime. In particular, in a short junction, a large anomalous current is a signature for topologically nontrivial superconductivity in the nanowire. Show less

Discrete energy levels of ultrasmall metallic grains are extracted in single-electron tunneling spectroscopy experiments. We study the response of these energy levels to an external magnetic field in the presence of both spin-orbit scattering and pairing correlations. In particular, we investigate g factors and level curvatures that parametrize, respectively, the linear and quadratic terms in the magnetic-field dependence of the many-particle energy levels of the grain. Both of these quantities… Show more

Discrete energy levels of ultrasmall metallic grains are extracted in single-electron tunneling spectroscopy experiments. We study the response of these energy levels to an external magnetic field in the presence of both spin-orbit scattering and pairing correlations. In particular, we investigate g factors and level curvatures that parametrize, respectively, the linear and quadratic terms in the magnetic-field dependence of the many-particle energy levels of the grain. Both of these quantities exhibit level-to-level fluctuations in the presence of spin-orbit scattering. We show that the distribution of g factors is not affected by the pairing interaction and that the distribution of level curvatures is sensitive to pairing correlations even in the smallest grains in which the pairing gap is smaller than the mean single-particle level spacing. We propose the level curvature in a magnetic field as a tool to probe pairing correlations in tunneling spectroscopy experiments. Show less

A nano-scale metallic grain (nanoparticle) with irregular boundaries in which the single-particle dynamics are chaotic is a zero-dimensional system described by the so-called universal Hamiltonian in the limit of a large number of electrons. The interaction part of this Hamiltonian includes a superconducting pairing term and a ferromagnetic exchange term. Spin-orbit scattering breaks spin symmetry and suppresses the exchange interaction term. Of particular interest is the fluctuation-dominated… Show more

A nano-scale metallic grain (nanoparticle) with irregular boundaries in which the single-particle dynamics are chaotic is a zero-dimensional system described by the so-called universal Hamiltonian in the limit of a large number of electrons. The interaction part of this Hamiltonian includes a superconducting pairing term and a ferromagnetic exchange term. Spin-orbit scattering breaks spin symmetry and suppresses the exchange interaction term. Of particular interest is the fluctuation-dominated regime, typical of the smallest grains in the experiments, in which the bulk pairing gap is comparable to or smaller than the single-particle mean-level spacing, and the Bardeen-Cooper-Schrieffer (BCS) mean-field theory of superconductivity is no longer valid. Here we study the crossover between the BCS and fluctuation-dominated regimes in two limits. In the absence of spin-orbit scattering, the pairing and exchange interaction terms compete with each other. We describe the signatures of this competition in thermodynamic observables, the heat capacity and spin susceptibility. In the presence of strong spin-orbit scattering, the exchange interaction term can be ignored. We discuss how the magnetic-field response of discrete energy levels in such a nanoparticle is affected by pairing correlations. We identify signatures of pairing correlations in this response, which are detectable even in the fluctuation-dominated regime. Show less

We study the mesoscopic fluctuations of thermodynamic observables in a nanosized metallic grain in which the single-particle dynamics are chaotic and the dimensionless Thouless conductance is large. Such a grain is modeled by the universal Hamiltonian describing the competition between exchange and pairing correlations. The exchange term is taken into account exactly by a spin-projection method, and the pairing term is treated in the static-path approximation together with small-amplitude… Show more

We study the mesoscopic fluctuations of thermodynamic observables in a nanosized metallic grain in which the single-particle dynamics are chaotic and the dimensionless Thouless conductance is large. Such a grain is modeled by the universal Hamiltonian describing the competition between exchange and pairing correlations. The exchange term is taken into account exactly by a spin-projection method, and the pairing term is treated in the static-path approximation together with small-amplitude quantal fluctuations around each static fluctuation of the pairing field. Odd-even particle-number effects induced by pairing correlations are included using a number-parity projection. We find that the exchange interaction shifts the number-parity effects in the heat capacity and spin susceptibility to lower temperatures. In the regime where the pairing gap is similar to or smaller than the single-particle mean level spacing, these number-parity effects are suppressed by exchange correlations, and the fluctuations of the spin susceptibility may be particularly large. However, for larger values of the pairing gap, the number-parity effects may be enhanced by exchange correlations. Show less

A nano-scale metallic grain in which the single-particle dynamics are chaotic is described by
the so-called universal Hamiltonian. This Hamiltonian includes a superconducting pairing
term and a ferromagnetic exchange term that compete with each other: pairing correlations
favor minimal ground-state spin, while the exchange interaction favors maximal spin
polarization. Of particular interest is the fluctuation-dominated regime where the bulk pairing
gap is comparable with or… Show more

A nano-scale metallic grain in which the single-particle dynamics are chaotic is described by
the so-called universal Hamiltonian. This Hamiltonian includes a superconducting pairing
term and a ferromagnetic exchange term that compete with each other: pairing correlations
favor minimal ground-state spin, while the exchange interaction favors maximal spin
polarization. Of particular interest is the fluctuation-dominated regime where the bulk pairing
gap is comparable with or smaller than the single-particle mean level spacing and the
Bardeen–Cooper–Schrieffer theory of superconductivity breaks down. Superconductivity and
ferromagnetism can coexist in this regime. We identify signatures of the competition between
superconductivity and ferromagnetism in a number of quantities: ground-state spin,
conductance fluctuations when the grain is weakly coupled to external leads and the
thermodynamic properties of the grain, such as heat capacity and spin susceptibility. Show less