Schemes for topological quantum computation with Majorana bound states rely heavily on the ability to measure products of Majorana operators projectively. Here, we employ Markovian quantum measurement theory, including the readout device, to analyze such measurements. Specifically, we focus on the readout of Majorana qubits via continuous charge sensing of a tunnel-coupled quantum dot by a quantum point contact. We show that projective measurements of Majorana products $\prod_i\hatγ_i$ can be implemented by continuous charge sensing under quite general circumstances. Essential requirements are that a combined local parity $\hatπ$, involving the quantum dot charge along with the Majorana product of interest, be conserved, and that the two eigenspaces of the combined parity $\hatπ$ generate distinguishable measurement signals. We find that qubit readout may have to rely on measuring noise correlations of the quantum-point-contact current. The average current encodes the qubit readout only transiently for fine-tuned parameters or in the presence of relaxation processes. We also discuss the corresponding measurement and decoherence times and consider processes such as residual Majorana hybridizations which are detrimental to the measurement protocol. Finally, we emphasize that the underlying mechanism -- which we term symmetry-protected readout -- is quite general and has further implications for both, Majorana and non-Majorana systems.
We analyze a readout scheme for Majorana qubits based on dispersive coupling to a resonator. We consider two variants of Majorana qubits: the Majorana transmon and the Majorana box qubit. In both cases, the qubit-resonator interaction can produce sizeable dispersive shifts in the MHz range for reasonable system parameters, allowing for submicrosecond readout with high fidelity. For Majorana transmons, the light-matter interaction used for readout manifestly conserves Majorana parity, which leads to a notion of quantum nondemolition (QND) readout that is stronger than for conventional charge qu…
We propose a quantum non-demolition Majorana qubit readout protocol based on parametricmodulation of a longitudinal interaction between a pair of Majorana bound states and a resonator. The interaction can be modulated through microwave frequency gate-voltage or flux control of a tunable tunnel barrier. The qubit-resonator coupling is quantum non-demolition to exponential accuracy, a property inherited from the topological nature of the Majorana zero modes. The same mechanism as used for single-qubit readout can also be extended to perform multi-qubit measurements and be used to enact long rang…
Solid-state qubits incorporating quantum dots can be read out by gate reflectometry. Here, we theoretically describe physical mechanisms that render such reflectometry-based readout schemes imperfect. We discuss charge qubits, singlet-triplet spin qubits, and Majorana qubits. In our model, we account for readout errors due to slow charge noise, and due to overdriving, when a too strong probe is causing errors. A key result is that for charge and spin qubits, the readout fidelity saturates at large probe strengths, whereas for Majorana qubits, there is an optimal probe strength which provides a…
We study the time-dependent effect of Markovian readout processes on Majorana qubits whose parity degrees of freedom are converted into the charge of a tunnel-coupled quantum dot. By applying a recently established effective Lindbladian approximation [1-3], we obtain a completely positive and trace preserving Lindblad master equation for the combined dot-qubit dynamics, describing relaxation and decoherence processes beyond the rotating-wave approximation. This approach is applicable to a wide range of weakly coupled environments representing experimentally relevant readout devices. We study i…
Reading out Majorana bound states (MBSs) is essential both to verify their non-Abelian property and to realize topological quantum computation. Here, we construct a protocol to measure the parity of two MBSs in a Majorana island coupled to double quantum dot (DQD). The parity information is mapped to the charge state of the DQD through Landau-Zener transition. The operation needed is sweeping the bias of the DQD, which is followed by charge sensing. In the case without fine-tuning, a single run of sweep-and-detection implement a weak measurement of the parity. We find that in general a sequenc…
Majorana qubits offer a promising way to store and manipulate quantum information by encoding it into the state of Majorana zero modes. As the information is stored in a topological property of the system, local noise cannot lead to decoherence. Manipulation of the information is achieved by braiding the zero modes. The measurement, however, is challenging as the information is well hidden and thus inherently hard to access. Here, we discuss a setup for measuring the state of a Majorana qubit by employing standard tools of microwave engineering. The basic physical effect that we employ is the …
We propose an Aharonov-Bohm interferometer consisted of two topological superconducting chains (TSCs) to generate coherence of Majorana qubits, each qubit is made of two Majorana zero modes (MZMs) with the definite fermion parity. We obtain the generalized exact master equation as well as its solution and study the real-time dynamics of the MZM qubit states under various operations. We demonstrate that by tuning the magnetic flux, the decoherence rates can be modified significantly, and dissipationless MZMs can be generated. By applying the bias voltage to the leads, one can manipulate MZM qub…
We study a readout scheme of superconducting flux qubit state with a Cooper pair box as a transmon. The qubit states consist of the superpositions of two degenerate states where the charge and phase degrees of freedom are entangled. Owing to the robustness of transmon against external fluctuations, our readout scheme enables the quantum non-demolition and single-shot measurement of flux qubit states. The qubit state readout can be performed by using the non-linear Josephson amplifiers after a $π/2$-rotation driven by an ac-electric field.
We propose a hybrid system composed of a Majorana qubit and a nanomechanical resonator, implemented by a spin-orbit-coupled superconducting nanowire, using a set of static and oscillating ferromagnetic gates. The ferromagnetic gates induce Majorana bound states in the nanowire, which hybridize and constitute a Majorana qubit. Due to the oscillation of one of these gates, the Majorana qubit can be coherently rotated. By tuning the gate voltage to modulate the local spin-orbit coupling, it is possible to reach the resonance of the qubit-oscillator system for relatively strong couplings.
We show how to realize high-fidelity quantum non-demolition qubit readout using longitudinal qubit-oscillator interaction. This is realized by modulating the longitudinal coupling at the cavity frequency. The qubit-oscillator interaction then acts as a qubit-state dependent drive on the cavity, a situation that is fundamentally different from the standard dispersive case. Single-mode squeezing can be exploited to exponentially increase the signal-to-noise ratio of this readout protocol. We present an implementation of this idea in circuit quantum electrodynamics and a possible multi-qubit arch…
Single-shot qubit readout typically combines high readout contrast with long-lived readout signals, leading to large signal-to-noise ratios and high readout fidelities. In recent years, it has been demonstrated that both readout contrast and readout signal lifetime, and thus the signal-to-noise ratio, can be enhanced by forcing the qubit state to transition through intermediate states. In this work, we demonstrate that the sub-Poissonian relaxation statistics introduced by intermediate states can reduce the single-shot readout error rate by orders of magnitude even when there is no increase in…
Determination of qubit initialisation and measurement fidelity is important for the overall performance of a quantum computer. However, the method by which it is calculated in semiconductor qubits varies between experiments. In this paper we present a full theoretical analysis of electronic single-shot readout and describe critical parameters to achieve high fidelity readout. In particular, we derive a model for energy selective state readout based on a charge detector response and examine how to optimise the fidelity by choosing correct experimental parameters. Although we focus on single ele…
Qubit realizations based on Majorana bound states have been considered promising candidates for quantum information processing which is inherently inert to decoherence. We put the underlying general arguments leading to this conjecture to the test from an open quantum system perspective. It turns out that, from a fundamental point of view, the Majorana qubit is as susceptible to decoherence as any local paradigm of a qubit.
We study the decoherence dynamics of Majorana qubit braiding operations in a topological superconducting chain (TSC) system, in which the braiding is performed by controlling the electron-chemical potentials of the TSCs and the couplings between them. By solving rigorously the Majorana qubit dynamics, we show how the Majorana qubit coherence is generated through bogoliubon correlations formed by exchanging Majorana zero modes (MZMs) between two TSCs in braiding operations. Using the exact master equation, we demonstrate how MZMs and also the bogoliubon correlations dissipate due to charge fluc…
We propose and demonstrate a frequency-multiplexed readout scheme in 3D cQED architecture. We use four transmon qubits coupled to individual rectangular cavities which are aperture-coupled to a common rectangular waveguide feedline. A coaxial to waveguide transformer at the other end of the feedline allows one to launch and collect the multiplexed signal. The reflected readout signal is amplified by an impedance engineered broadband parametric amplifier with 380 MHz of bandwidth. This provides us high fidelity single-shot readout of multiple qubits using compact microwave circuitry, an efficie…
In dispersive readout schemes, qubit-induced nonlinearity typically limits the measurement fidelity by reducing the signal-to-noise ratio (SNR) when the measurement power is increased. Contrary to seeing the nonlinearity as a problem, here we propose to use it to our advantage in a regime where it can increase the SNR. We show analytically that such a regime exists if the qubit has a many-level structure. We also show how this physics can account for the high-fidelity avalanchelike measurement recently reported by Reed {\it et al.} [arXiv:1004.4323v1].
Majorana zero modes have been attracting considerable attention because of their prospective applications in fault-tolerant topological quantum computing. In recent years, some schemes have been proposed to detect and manipulate Majorana zero modes using superconducting qubits. However, manipulating and reading the Majorana zero modes must be kept in the time window of quasiparticle poisoning. In this work, we study the problem of quasiparticle poisoning in a split transmon qubit containing hybrid Josephson junctions involving Majorana zero modes. We show that Majorana coupling will cause pari…
Majorana bound states have been a focus of condensed matter research for their potential applications in topological quantum computation. Here we utilize two charge-qubit arrays to explicitly simulate a DIII class one-dimensional superconductor model where Majorana end states can appear. Combined with one braiding operation, universal single-qubit operations on a Majorana-based qubit can be implemented by a controllable inductive coupling between two charge qubits at the ends of the arrays. We further show that in a similar way, a controlled-NOT gate for two topological qubits can be simulated…
Quantum systems must be prepared, controlled, and measured with high fidelity in order to perform complex quantum algorithms. Control fidelities have greatly improved in silicon spin qubits, but state preparation and readout fidelities have generally been poor. By operating with low electron temperatures and employing high-bandwidth cryogenic amplifiers, we demonstrate single qubit readout visibilities >99%, exceeding the threshold for quantum error correction. In the same device, we achieve average single qubit control fidelities >99.95%. Our results show that silicon spin qubits can be opera…
A primary method of reading out a quantum dot hybrid qubit involves projection of the logical basis onto distinct charge states that are readily detected by an integrated charge sensing dot. However, in the simplest configuration, the excited charge state decays rapidly, making single-shot readout challenging. Here, we demonstrate a readout procedure where the qubit excited state is latched to a metastable charge configuration whose lifetime is tunnel rate limited, persisting here as long as 2.5 ms. Additionally, we show that working in the (4,1)-(3,2) charge configuration enables a latched re…
We present a theoretical analysis of a Majorana-based qubit consisting of two topological superconducting islands connected via a Josephson junction. The qubit is operated by electrostatic gates which control the coupling of two of the four Majorana zero modes. At the end of the operation, readout is performed in the charge basis. Even though the operations are not topologically protected, the proposed experiment can potentially shed light on the coherence of the parity degree of freedom in Majorana devices and serve as a first step towards topological Majorana qubits. We discuss in detail the…
We theoretically examine a scheme for projectively reading out the parity state of a pair of Majorana bound states (MBS) using a tunnel coupled quantum dot. The dot is coupled to one end of the topological wire but isolated from any reservoir, and is capacitively coupled to a charge sensor for measurement. The combined parity of the MBS-dot system is conserved and charge transfer between the MBS and dot only occurs through resonant tunnelling. Resonance is controlled by the dot potential through a local gate and by the MBS energy splitting due to the overlap of the MBS pair wavefunctions. The …
We study bifurcation measurement of a multi-level superconducting qubit using a nonlinear resonator biased in the straddling regime, where the resonator frequency sits between two qubit transition frequencies. We find that high-fidelity bifurcation measurements are possible because of the enhanced qubit-state-dependent pull of the resonator frequency, the behavior of qubit-induced nonlinearities and the reduced Purcell decay rate of the qubit that can be realized in this regime. Numerical simulations find up to a threefold improvement in qubit readout fidelity when operating in, rather than ou…
We study qubit decoherence under generalized dispersive readout, i.e., we investigate a qubit coupled to a resonantly driven dissipative harmonic oscillator. We provide a complete picture by allowing for arbitrarily large qubit-oscillator detuning and by considering also a coupling to the square of the oscillator coordinate, which is relevant for flux qubits. Analytical results for the decoherence time are obtained by a transformation of the qubit-oscillator Hamiltonian to the dispersive frame and a subsequent master equation treatment beyond the Markov limit. We predict a crossover from Marko…
Qubit readout is commonly performed by thresholding a collection of analog detector signals to obtain a sequence of single-shot bit values. The intrinsic irreversibility of the mapping from analog to digital signals discards soft information associated with an \emph{a posteriori} confidence that can be assigned to each bit value when a detector is well characterized. Accounting for soft information, we show significant improvements in enhanced state detection with the quantum repetition code as well as quantum state or parameter estimation. These advantages persist in spite of non-Gaussian fea…
Although Majorana platforms are promising avenues to realizing topological quantum computing, they are still susceptible to errors from thermal noise and other sources. We show that the error rate of Majorana qubits can be drastically reduced using a 1D repetition code. The success of the code is due the imbalance between the phase error rate and the flip error rate. We demonstrate how a repetition code can be naturally constructed from segments of Majorana nanowires. We find the optimal lifetime may be extended from a millisecond to over one second.
Reading out the parity degree of freedom of Majorana bound states is key to demonstrating their nonabelian exchange properties. Here, we present a low-energy model describing localized edge states in a two-arm device. We study parity-to-charge conversion based on coupling the superconductor bound states to a quantum dot whose charge is read out by a sensor. The dynamics of the system, including the readout device, is analyzed in full using a quantum-jump approach. We show how the resulting signal and noise ratio differentiates between local Majorana and Andreev bound states.
A better understanding of the noise causing qubit decoherence is crucial for improving qubit performance. The noise spectrum affecting the qubit may be extracted by measuring dephasing under the application of pulse sequences but requires accurate qubit control and sufficiently long relaxation times, which are not always available. Here, we describe an alternative method to extract the spectrum from correlations of single-shot measurement outcomes of successive free induction decays. This method only requires qubit initialization and readout with a moderate fidelity and also allows independent…
We construct a minimal circuit, based on the top-transmon design, to rotate a qubit formed out of four Majorana zero-modes at the edge of a two-dimensional topological insulator. Unlike braiding operations, generic rotations have no topological protection, but they do allow for a full characterization of the coherence times of the Majorana qubit. The rotation is controlled by variation of the flux through a pair of split Josephson junctions in a Cooper pair box, without any need to adjust gate voltages. The Rabi oscillations of the Majorana qubit can be monitored via oscillations in the resona…
An important desired ingredient of superconducting quantum circuits is a readout scheme whose complexity does not increase with the number of qubits involved in the measurement. Here, we present a readout scheme employing a single microwave line, which enables simultaneous readout of multiple qubits. Consequently, scaling up superconducting qubit circuits is no longer limited by the readout apparatus. Parallel readout of 6 flux qubits using a frequency division multiplexing technique is demonstrated, as well as simultaneous manipulation and time resolved measurement of 3 qubits. We discuss how…
We present a superconducting device that realizes the sequential measurement of a transmon qubit. The device disables common limitations of dispersive readout such as Purcell effect or transients in the cavity mode by turning on and off the coupling to the measurement channel on demand. The qubit measurement begins by loading a readout resonator that is coupled to the qubit. After an optimal interaction time with negligible loss, a microwave pump releases the content of the readout mode by upconversion into a measurement line in a characteristic time as low as 10~ns, which is 400 times shorter…
We introduce and analyze a dispersive qubit readout scheme where two-mode squeezing is generated directly in the measurement cavities. The resulting suppression of noise enables fast, high- fidelity readout of naturally weakly coupled qubits, and the possibility to protect strongly coupled qubits from decoherence by weakening their coupling. Unlike other approaches exploiting squeezing, our setup avoids the difficult task of transporting and injecting with high fidelity an externally-generated squeezed state. Our setup is also surprisingly robust against unwanted non-QND backaction effects, as…
The dispersive coupling between a qubit and a cavity mode is widely used for performing non-destructive readout of the qubit state. In this approach, it is typically required that the dispersive strong coupling regime is achieved. Here we show that the use of an auxiliary cavity mode reduces by orders of magnitude the required value of the dispersive coupling, for a given decay rate of the cavity mode. The analysis is performed within the input-output formalism, in terms of the photon scattering matrix elements and of the signal-to-noise ratio. We derive simple analytical expressions for the o…
Quantum bits based on Majorana zero modes are expected to be robust against certain noise types, and hence provide a quantum computing platform that is superior to conventional qubits. This robustness is not complete though: imperfections can still lead to qubit decoherence and hence to information loss. In this work, we theoretically study Majorana-qubit dephasing in a minimal model: in a Kitaev chain with quasistatic disorder. Our approach, based on numerics as well as first-order non-degenerate perturbation theory, provides a conceptually simple physical picture and predicts Gaussian dephas…
A Majorana-based qubit can form within the fermion parity subspace of a two-island topological superconductor setup consisting of four Majorana zero modes. We study the relaxation time of this Majorana qubit induced by the intrinsic Majorana-phonon interaction in combination with single electron tunneling between the two islands. The theory is developed for both the spinless p-wave Kitaev chain model and the spin-orbit-coupled semiconductor nanowire with induced superconductivity and Zeeman splitting. We systematically discuss the dependence of the relaxation rate on Majorana charge distributi…
Machine Learning finds application in the quantum control and readout of qubits. In this work we apply Artificial Neural Networks to assist the manipulation and the readout of a prototypical molecular spin qubit - an Oxovanadium(IV) moiety - in two experiments designed to test the amplitude and the phase recognition, respectively. We first successfully use an artificial network to analyze the output of a Storage/Retrieval protocol with four input pulses to recognize the echo positions and, with further post selection on the results, to infer the initial input pulse sequence. We then apply an A…
Motivated by recent proposals of Majorana qubits and the read-out of their quantum state we investigate a qubit setup formed by two parallel topological wires shunted by a superconducting bridge. The wires are further coupled to two quantum dots, which are also linked directly, thus creating an interference loop. The transport current through this system shows an interference pattern which distinguishes two basis states of the qubit in a QND measurement. We analyze various properties of the interference current and the read-out process, including the resulting dephasing and relaxation. We also…
Topological quantum computation encodes quantum information nonlocally by nucleating non-Abelian anyons separated by distances $L$, typically spanning the qubit device size. This nonlocality renders topological qubits exponentially immune to dephasing from all sources of classical noise with operator support local on the scale of $L$. We perform detailed analytical and numerical analyses of a time-domain Ramsey-type protocol for noisy Majorana-based qubits that is designed to validate this coveted topological protection in near-term devices such as the so-called `tetron' design. By assessing d…
We introduce a frequency-multiplexed readout scheme for superconducting phase qubits. Using a quantum circuit with four phase qubits, we couple each qubit to a separate lumped-element superconducting readout resonator, with the readout resonators connected in parallel to a single measurement line. The readout resonators and control electronics are designed so that all four qubits can be read out simultaneously using frequency multiplexing on the one measurement line. This technology provides a highly efficient and compact means for reading out multiple qubits, a significant advantage for scali…
We theoretically study a scheme to distinguish the two ground states of a one-dimensional topological superconductor, which could serve as a basis for the readout of Majorana qubits. The scheme is based on parity-to-charge conversion, i.e., the ground-state parity of the superconductor is converted to the charge occupation on a tunnel-coupled auxiliary quantum dot. We describe how certain error mechanisms degrade the quality of the parity-to-charge conversion process. We consider (i) leakage due to a strong readout tunnel pulse, (ii) incomplete charge Rabi oscillations due to slow charge noise…
We report single-shot readout of a superconducting flux qubit by using a flux-driven Josephson parametric amplifier (JPA). After optimizing the readout power, gain of the JPA and timing of the data acquisition, we observe the Rabi oscillations with a contrast of 74% which is mainly limited by the bandwidth of the JPA and the energy relaxation of the qubit. The observation of quantum jumps between the qubit eigenstates under continuous monitoring indicates the nondestructiveness of the readout scheme.
We present a new readout method for a superconducting flux qubit, based on the measurement of the Josephson inductance of a superconducting quantum interference device that is inductively coupled to the qubit. The intrinsic flux detection efficiency and back-action are suitable for a fast and nondestructive determination of the quantum state of the qubit, as needed for readout of multiple qubits in a quantum computer. We performed spectroscopy of a flux qubit and we measured relaxation times of the order of 80 $μs$.
High fidelity quantum information processing requires a combination of fast gates and long-lived quantum memories. In this work, we propose a hybrid architecture, where a parity-protected superconducting qubit is directly coupled to a Majorana qubit, which plays the role of a quantum memory. The superconducting qubit is based upon a $π$-periodic Josephson junction realized with gate-tunable semiconducting wires, where the tunneling of individual Cooper pairs is suppressed. One of the wires additionally contains four Majorana zero modes that define a qubit. We demonstrate that this enables the …
We present a method for measuring the internal state of a superconducting qubit inside an on-chip microwave resonator. We show that one qubit state can be associated with the generation of an increasingly large cavity coherent field, while the other remains associated with the vacuum. By measuring the outgoing resonator field with conventional devices, an efficient single-shot QND-like qubit readout can be achieved, enabling a high-fidelity measurement in the spirit of the electron-shelving technique for trapped ions. We expect that the proposed ideas can be adapted to different superconductin…
The CNOT gate is a two-qubit gate which is essential for universal quantum computation. A well-established approach to implement it within Majorana-based qubits relies on subsequent measurement of (joint) Majorana parities. We propose an alternative scheme which operates a protected CNOT gate via the holonomic control of a handful of system parameters, without requiring any measurement. We show how the adiabatic tuning of pair-wise couplings between Majoranas can robustly lead to the full entanglement of two qubits, insensitive with respect to small variations in the control of the parameters.
We have developed a cryogenic platform for the control and readout of spin qubits that comprises a high density of dc and radio frequency sample interconnects based on a set of coupled printed circuit boards. The modular setup incorporates 24 filtered dc lines, 14 control and readout lines with bandwidth from dc to above 6 GHz, and 2 microwave connections for excitation to 40 GHz. We report the performance of this platform, including signal integrity and crosstalk measurements and discuss design criteria for constructing sample interconnect technology needed for multi-qubit devices.
We investigate effects of the particle-hole asymmetry on the properties of Majorana qubtis. Particle-hole asymmetric pertrubation shifts Majorana zero mode from zero-energy. This type of asymmetry can act as a source of decoherence in Majorana qubits. We find that particle-hole asymmetric pertrubation causes phase-shifts and bit-flips decoherence processes in Majorana qubits. However, these sources of decoherence are negligible in uniform system with spatially separated Majorana zero modes.
We propose a platform for universal quantum computation that uses conventional $s$-wave superconducting leads to address a topological qubit stored in spatially separated Majorana bound states in a multi-terminal topological superconductor island. Both the manipulation and read-out of this "Majorana superconducting qubit" are realized by tunnel couplings between Majorana bound states and the superconducting leads. The ability of turning on and off tunnel couplings on-demand by local gates enables individual qubit addressability while avoiding cross-talk errors. By combining the scalability of …
We propose a scheme to implement Controlled Not-gate for topological qubits in a quantum-dot-Majorana-fermion hybrid system. Quantum information is encoded on pairs of Majorana fermions, which live on the the interface between topologically trivial and nontrivial sections of a quantum nanowire deposited on an s-wave superconductor. A measurement-based two-qubit Controlled-Not gate is produced with the help of parity measurements assisted by the quantum-dot and followed by prescribed single-qubit gates. The parity measurement, on the quantum-dot and a topological qubit, is achieved by the Aharo…
