Inevitably, assessing the overall performance of a quantum computer must rely on characterizing some of its elementary constituents and, from this information, formulate a broader statement concerning more complex constructions thereof. However, given the vastitude of possible quantum errors as well as their coherent nature, accurately inferring the quality of composite operations is generally difficult. To navigate through this jumble, we introduce a non-physical simplification of quantum maps that we refer to as the leading Kraus (LK) approximation. The uncluttered parameterization of LK approximated maps naturally suggests the introduction of a unitary-decoherent polar factorization for quantum channels in any dimension. We then leverage this structural dichotomy to bound the evolution -- as circuits grow in depth -- of two of the most experimentally relevant figures of merit, namely the average process fidelity and the unitarity. We demonstrate that the leeway in the behavior of the process fidelity is essentially taken into account by physical unitary operations.
Inevitably, assessing the overall performance of a quantum computer must rely on characterizing some of its elementary constituents and, from this information, formulate a broader statement concerning more complex constructions thereof. However, given the vastitude of possible quantum errors as well as their coherent nature, accurately inferring the quality of composite operations is generally difficult. To navigate through this jumble, we introduce a non-physical simplification of quantum maps that we refer to as the leading Kraus (LK) approximation. The uncluttered parameterization of LK approximated maps naturally suggests the introduction of a unitary-decoherent polar factorization for quantum channels in any dimension. We then leverage this structural dichotomy to bound the evolution -- as circuits grow in depth -- of two of the most experimentally relevant figures of merit, namely the average process fidelity and the unitarity. We demonstrate that the leeway in the behavior of the process fidelity is essentially taken into account by physical unitary operations.
Present-day quantum processors are open systems in which dissipation and decoherence degrade the information carried by a quantum state as it propagates through a circuit. We study this degradation in dissipative random quantum circuits, modeling each two-qubit gate as a diluted unitary that interpolates between the intended unitary operation and a random dissipative quantum channel. Using the fidelity between the ideal and noisy output states as a diagnostic, we show that sufficiently random unitary gates or dissipative Kraus operators lead to a universal decay of the average fidelity. This decay is determined only by the dissipation strength, system size, and circuit depth, while microscopic details of the gates and noise, including the Kraus rank, appear only in subleading corrections to higher moments. We derive a closed-form expression for the average fidelity in terms of three physically meaningful error parameters, dissipation strength, coherent two-qubit gate-error strength, and connectivity-error probability. Finally, we identify the cases when dissipative effects can be reproduced by an unitary-noise model and when dissipation-induced decoherence remains distinguishable from coherent noise at the level of the average fidelity.
Every potential benefit of quantum computers would be lost without methods to protect the computations from error. Quantum channels provide a framework for understanding error in quantum systems. In general, for a system of size d, a quantum channel may require as many as ~d^4 parameters to characterize it. We introduce the leading Kraus approximation, a simplification which reduces the number of parameters to ~d^2, while accurately approximating two figures of merit important to the experimentalist: the unitarity and average process fidelity. Additionally, applying the leading Kraus approximation declutters investigations into the set of quantum channels. When applied, a natural decomposition of channels arises, separating behaviour into coherent and decoherent contributions. We find that eliminating the coherent contribution provides the greatest increase to the fidelity, while decoherent processes provide a generalization of depolarizing channels.
The problem of an optimal mapping between Hilbert spaces IN and OUT, based on a series of density matrix mapping measurements ${\ensuremath{\rho}}^{(l)}\ensuremath{\rightarrow}{\ensuremath{\varrho}}^{(l)}$, $l=1\ensuremath{\cdots}M$, is formulated as an optimization problem maximizing the total fidelity $\mathcal{F}={\ensuremath{\sum}}_{l=1}^{M}{\ensuremath{\omega}}^{(l)}F({\ensuremath{\varrho}}^{(l)},{\ensuremath{\sum}}_{s}{B}_{s}{\ensuremath{\rho}}^{(l)}{B}_{s}^{\ifmmode\dagger\else\textdagger\fi{}})$ subject to probability preservation constraints on Kraus operators Bs. For $F(\ensuremath{\varrho},\ensuremath{\sigma})$ in the form that total fidelity can be represented as a quadratic form with superoperator $\mathcal{F}={\ensuremath{\sum}}_{s}\ensuremath{\langle}{B}_{s}|S|{B}_{s}\ensuremath{\rangle}$ (either exactly or as an approximation) an iterative algorithm is developed. The work introduces two important generalizations of unitary learning: (1) IN/OUT states are represented as density matrices; (2) the mapping itself is formulated as a mixed unitary quantum channel ${A}^{OUT}={\ensuremath{\sum}}_{s}{|{w}_{s}|}^{2}{\mathcal{U}}_{s}{A}^{IN}{\mathcal{U}}_{s}^{\ifmmode\dagger\else\textdagger\fi{}}$ (no general quantum channel yet). This marks a crucial advancement from the commonly studied unitary mapping of pure states ${\ensuremath{\phi}}_{l}=\mathcal{U}{\ensuremath{\psi}}_{l}$ to a quantum channel, which allows us to distinguish probabilistic mixture of states and their superposition. An application of the approach is demonstrated on unitary learning of density matrix mapping ${\ensuremath{\varrho}}^{(l)}=\mathcal{U}{\ensuremath{\rho}}^{(l)}{\mathcal{U}}^{\ifmmode\dagger\else\textdagger\fi{}}$, in this case a quadratic on U fidelity can be constructed by considering $\sqrt{{\ensuremath{\rho}}^{(l)}}\ensuremath{\rightarrow}\sqrt{{\ensuremath{\varrho}}^{(l)}}$ mapping, and on a quantum channel, where quadratic on Bs fidelity is an approximation---a quantum channel is then obtained as a hierarchy of unitary mappings, a mixed unitary channel. The approach can be applied to studying quantum inverse problems, variational quantum algorithms, quantum tomography, and more. A software product implementing the algorithm is available from the authors.
Approximate quantum error correction (AQEC) extends the framework of discrete- and continuous-variable quantum error correction beyond the Knill-Laflamme (KL) conditions, where the recovery performance is quantified by entanglement fidelity. Recent studies have enabled efficient evaluation of near-optimal entanglement fidelity using transpose-channel recovery. Yet, determining the global optimal recovery map and its entanglement fidelity for general codes beyond the KL conditions remains a major computational challenge. Direct optimization becomes prohibitive as the number of noise Kraus operators grows rapidly with system size, and existing approaches lack rigorous guarantees for reducing this optimization to a tractable dimension. Here, we derive an explicit characterization of the optimal environmental state of complement channel, which transforms the optimization over recovery channels into an equivalent optimization over quotient unitaries. For a broader class of codes that satisfy only the orthogonality part of the KL conditions, we show that the optimal recovery map admits an explicit analytical form. Building on this form, we derive novel rigorous lower bounds of entanglement fidelity that strictly improve upon the transpose-recovery bound. We further develop a novel recovery strategy based on principle components, and derive a rigorous bound on the error introduced by noise truncation. Our approach enables efficient searches for approximate recovery maps for AQEC codes, avoiding the need to optimize over the full Kraus-operator space.
In the study of quantum computation, data is represented in terms of linear operators which form a generalized model of probability, and computations are most commonly described as products of unitary transformations, which are the transformations which preserve the quality of the data in a precise sense. This naturally leads to "unitary circuit models", which are models of computation in which unitary operators are expressed as a product of "elementary" unitary transformations. However, unitary transformations can also be effected as a composition of operations which are not all unitary themselves: the "one-way measurement model" is one such model of quantum computation. In this thesis, we examine the relationship between representations of unitary operators and decompositions of those operators in the one-way measurement model. In particular, we consider different circumstances under which a procedure in the one-way measurement model can be described as simulating a unitary circuit, by considering the combinatorial structures which are common to unitary circuits and two simple constructions of one-way based procedures. These structures lead to a characterization of the one-way measurement patterns which arise from these constructions, which can then be related to efficiently testable properties of graphs. We also consider how these characterizations provide automatic techniques for obtaining complete measurement-based decompositions, from unitary transformations which are specified by operator expressions bearing a formal resemblance to path integrals. These techniques are presented as a possible means to devise new algorithms in the one-way measurement model, independently of algorithms in the unitary circuit model.
Quantum computers have the potential to bring about a new age of technology, but in order for them to become a reality, the elephant in the room must be addressed, quantum errors. The complexity of quantum errors grows exponentially as the system size is increased; scalable characterization protocols which work under physically realistic noise are therefore needed to assess and optimize control. \n \nRandomized Benchmarking (RB) is a family of characterization protocols with the requisite qualities. RB protocols are independent of state preparation and measurement (SPAM) errors, and because they extract partial information about the error profiles in the form of figures of merit, they are scalable in the number of qubits. In this thesis, we study these figures of merit under various error models, gate-sets, and sample distributions, to further our understanding of their physical meaning, and to fine-tune the best practice for diagnosing errors in basic gate-sets. \n \nThe first study presented in this thesis looks at the physical meaning of the unitarity under physically realistic gate-dependent noise. Numerically, we demonstrate that under these realistic noise models, the unitarity obtained via Extended RB (XRB) overlaps strongly with the average of the unitarity of the errors over the set of individual operations. \n \nThe second study numerically and analytically proves that the Standard RB (SRB) decay parameter p coincides with the decay parameter of the novel figure of merit gate-set circuit fidelity. We also prove that, in the limit of high-fidelity single-qubit experiments, the disconnect between the average gate fidelity and SRB experimental results is caused by a basis mismatch between the gates and the state-preparation and measurement procedures. \n \nIn the third part of this thesis, we compare the results of two variations of the SRB protocol, one that samples uniformly (SRB) and one that samples non-uniformly (NIST RB) from the Clifford group. We prove that NIST RB produces an exponential fidelity curve that gives different infidelity results than SRB. \n \nThe fourth and final study gives a standard for assessing basic gate-sets. Experimental results are given from the assessment run on super-conducting and ion-trap devices. We also experimentally show how the systematic error bounds can be tightened by accounting for the error's degree of coherence, as well as using weaker randomizing sets.