The measurement of a quantum two-level system, or a qubit in modern terminology, often involves an electromagnetic field that interacts with the qubit, before the field is measured continuously and the qubit state is inferred from the noisy field measurement. During the measurement, the qubit may undergo spontaneous transitions, further obscuring the initial qubit state from the observer. Taking advantage of some well-known techniques in stochastic detection theory, here we propose a signal processing protocol that can infer the initial qubit state optimally from the measurement in the presence of noise and qubit dynamics. Assuming continuous quantum-nondemolition measurements with Gaussian or Poissonian noise and a classical Markov model for the qubit, we derive analytic solutions to the protocol in some special cases of interest using It\ifmmode \bar{o}\else ={o}\fi{} calculus. Our method is applicable to multihypothesis testing for robust qubit readout and relevant to experiments on qubits in superconducting microwave circuits, trapped ions, nitrogen-vacancy centers in diamond, semiconductor quantum dots, or phosphorus donors in silicon.
We study how the spontaneous relaxation of a qubit affects a continuous quantum nondemolition measurement of the initial state of the qubit. Given some noisy measurement record $\ensuremath{\Psi}$, we seek an estimate of whether the qubit was initially in the ground or excited state. We investigate four different measurement protocols, three of which use a linear filter (with different weighting factors) and a fourth which uses a full nonlinear filter that gives the theoretically optimal estimate of the initial state of the qubit. We find that relaxation of the qubit at rate
1∕T1 strongly influences the fidelity of any measurement protocol. To avoid errors due to this decay, the measurement must be completed in a time that decrease linearly with the desired fidelity while maintaining an adequate signal to noise ratio. We find that for the nonlinear filter the predicted fidelity, as expected, is always better than the linear filters and that the fidelity is a monotone increasing function of the measurement time. For example, to achieve a fidelity of 90%, the box car linear filter requires a signal to noise ratio of $\ensuremath{\sim}30$ in a time
T1, whereas the nonlinear filter only requires a signal to noise ratio of $\ensuremath{\sim}18$.
For several decades it has been appreciated that quantum computers hold incredible promise to perform calculations intractable to classical computation. However, this promise has be slow to realize. Dozens of quantum systems are currently being investigated for use in quantum information processing - none of which have yet demonstrated algorithms involving more than a handful of qubits and it remains unclear which, if any, of these systems will ultimately compose a scalable, robust quantum information processing architecture. In this thesis we employ analytical, optimal and algebraic control techniques to evaluate various quantum systems for their potential use in quantum information processing. In doing so, we have additionally identified several novel characterization procedures capable of probing both the coherent and incoherent dynamics of quantum systems. The first part of this thesis discusses work motivated by attempts to utilize donor qubits in silicon as quantum bits. We first propose a measurement of the state of a single donor electron spin using two-dimensional electron gas of a field-effect transistor and electrically detected magnetic resonance. We analyze the potential sensitivity of this measurement and show that it is a quantum nondemolition measurement of an electron-encoded state. We then present the first of two novel qubit characterization procedures. We consider the problem of rapidly characterizing a large number of similarly prepared qubits using techniques from optimal experiment design. All qubits are assumed to evolve according to the same physical processes, though the Hamiltonian parameters may vary from device to device - an inevitability in solid state qubits. We use the Cram\\'er-Rao bound on the variance of a point estimator to construct the optimal series of experiments to estimate these free parameters, and present a complete analysis of the optimal experimental configuration. Though applied to dipole- and exchange-coupled qubits, this technique is widely applicable to other systems. The second part of the thesis discusses the role that control can play in measuring and mitigating noise in qubit systems. Our first result describes a method for quickly simulating the effects of arbitrary markovian noise on qubit systems through the use of a numerically optimized, multi-state Markovian fluctuator. This ability to rapidly simulate the noisy qubit evolution allows us to compute control sequences capable of maximally decoupling the qubit from the noise source. We then introduce the second characterization procedure of the these, showing that a single measurable and controllable qubit may act as a spectrometer of dephasing noise. We show that the formalism of dynamical decoupling can be used to estimate the short-time correlation function of the noise source, while long time correlations may be estimated by a very simple series of free evolution experiments. This technique is applicable to the wide range of physical implementations which suffer from dephasing noise.The final part of this thesis demonstrates that trapped neutral atoms may be utilized for the robust simulation of complex systems exhibiting a topological phase. We present a method to simulate the toric code Hamiltonian stroboscopically, and demonstrate that our technique preserves the ground state degeneracy . Furthermore, we introduce a dissipative mechanism allowing for thermalization of the system to a finite temperature or direct cooling to the ground state manifold.
We analyze three different post-processing methods applied to a single-shot qubit readout: the average-signal (boxcar filter), peak-signal, and maximum-likelihood methods. In contrast to previous work, we account for a stochastic turn-on time
ti associated with the leading edge of a pulse signaling one of the qubit states. This model is relevant to spin-qubit readouts based on spin-to-charge conversion and would be generically reached in the limit of large signal-to-noise ratio
r for several other physical systems, including fluorescence-based readouts of ion-trap qubits and nitrogen-vacancy center spins. We derive analytical closed-form expressions for the conditional probability distributions associated with the peak-signal and boxcar filters. For the boxcar filter, we find an asymptotic scaling of the single-shot error rate $\ensuremath{\varepsilon}\ensuremath{\sim}lnr/\sqrt{r}$ when
ti is stochastic, in contrast to the result $\ensuremath{\varepsilon}\ensuremath{\sim}lnr/r$ for deterministic
ti. Consequently, the peak-signal method outperforms the boxcar filter significantly when
ti is stochastic, but is only marginally better for deterministic
ti (a result that is consistent with the widespread use of the boxcar filter for fluorescence-based readouts and the peak signal for spin-to-charge conversion). We generalize the theoretically optimal maximum-likelihood method to stochastic
ti and show numerically that a stochastic turn-on time
ti will always result in a larger single-shot error rate. Based on this observation, we propose a general strategy to improve the quality of single-shot readouts by forcing
ti to be deterministic.
We propose a straightforward experimental protocol to test whether qubit-environment entanglement is generated during pure dephasing of a qubit. The protocol is implemented using only measurements and operations on the qubit---it does not involve the measurement of the system-environment state of interest, but the preparation and measurement of the qubit in two simple variations. A difference in the time dependencies of qubit coherence between the two cases testifies to the presence of entanglement in the state of interest. Furthermore, it signifies that the environment-induced noise experienced by the qubit cannot be modeled as a classical stochastic process independent of the qubit state. We demonstrate the operation of this protocol on a realistically modeled nitrogen-vacancy center spin qubit in diamond interacting with a nuclear spin environment, and show that the generation of entanglement should be easily observable in this case.
Abstract Quantum measurements are crucial for extracting information from quantum systems, but they are error-prone due to hardware imperfections in near-term devices. Measurement errors can be mitigated through classical post-processing, based on the assumption of a classical noise model. However, the coherence of quantum measurements leads to unavoidable quantum noise that defies this assumption. In this work, we introduce a two-stage procedure to systematically tackle such quantum noise in measurements. The idea is intuitive: we first detect and then eliminate quantum noise. In the first stage, inspired by coherence witness in the resource theory of quantum coherence, we design an efficient method to detect quantum noise. It works by fitting the difference between two measurement statistics to the Fourier series, where the statistics are obtained using maximally coherent states with relative phase and maximally mixed states as inputs. The fitting coefficients quantitatively benchmark quantum noise. In the second stage, we design various methods to eliminate quantum noise, inspired by the Pauli twirling technique. They work by executing randomly sampled Pauli gates before the measurement device and conditionally flipping the measurement outcomes in such a way that the effective measurement device contains only classical noise. We numerically demonstrate the two-stage procedure’s feasibility on the Baidu Quantum Platform. Notably, the results reveal significant suppression of quantum noise in measurement devices and substantial enhancement in quantum computation accuracy. We highlight that the two-stage procedure complements existing measurement error mitigation techniques, and they together form a standard toolbox for manipulating measurement errors in near-term quantum devices.
Atomic scale qubits, as may be realized in nitrogen vacancy (NV) centers in diamond, offer the opportunity to study magnetic field noise with nanometer scale spatial resolution. Using these spin qubits, one can learn a great deal about the magnetic-field noise correlations, and correspondingly the collective-mode spectra, in quantum materials and devices. However, to date these tools have been essentially restricted to studying Gaussian noise processes – equivalent to linear-response. In this work we will show how to extend these techniques beyond the Gaussian regime and show how to unambiguously measure higher-order magnetic noise cumulants in a local, spatially resolved way. We unveil two protocols for doing this; the first uses a single spin-qubit and different dynamical decoupling sequences to extract non-Markovian and non-Gaussian spin-echo noise. The second protocol uses two-qubit coincidence measurements to study spatially non-local cumulants in the magnetic noise. We then demonstrate the utility of these protocols by considering a model of a bath of non-interacting two-level systems, as well as a model involving spatially correlated magnetic fluctuations near a second-order Ising phase transition. In both cases, we highlight how this technique can be used to measure in a real many-body system how fluctuation dynamics converge towards the central limit theorem as a function of effective bath size. We then conclude by discussing some promising applications and extensions of this method.