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We study quasi-exact quantum error correcting codes and quantum computation with them. A quasi-exact code is an approximate code such that it contains a finite number of scaling parameters, the tuning of which can flow it to corresponding exact codes, serving as its fixed points. The computation with a quasi-exact code cannot realize any logical gate to arbitrary accuracy. To overcome this, the notion of quasi-exact universality is proposed, which makes quasi-exact quantum computation a feasible model especially for executing moderate-size algorithms. We find that the incompatibility between universality and transversality of the set of logical gates does not persist in the quasi-exact scenario. A class of covariant quasi-exact codes is defined which proves to support transversal and quasi-exact universal set of logical gates for $SU(d)$. This work opens the possibility of quantum computation with quasi-exact universality, transversality, and fault tolerance.
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Topological quantum computation started as a niche area of research aimed at employing particles with exotic statistics, called anyons, for performing quantum computation. Soon it evolved to include a wide variety of disciplines. Advances in the understanding of anyon properties inspired new quantum algorithms and helped in the characterisation of topological phases of matter and their experimental realisation. The conceptual appeal of topological systems as well as their promise for building fault-tolerant quantum technologies fuelled the fascination in this field. This `focus on brings together several of the latest developments in the field and facilitates the synergy between different approaches.
Despite free-fermion systems being dubbed exactly solvable, they generically do not admit closed expressions for nonlocal quantities such as topological string correlations or measures of entanglement. We derive closed expressions for such nonlocal quantities for a dense subclass of certain classes of topological fermionic wires (classes BDI and AIII). Our results also apply to spin chains known as generalised cluster models. While there is a bijection between general models in these fermionic classes and Laurent polynomials, restricting to polynomials with degenerate zeros leads to a plethora of exact results. In particular, (1) we derive closed expressions for the string correlation functions -- the order parameters for the topological phases in these classes; (2) we obtain an exact formula for the characteristic polynomial of the correlation matrix, giving insight into the entanglement of the ground state; (3) the latter implies that the ground state can be described by a matrix product state (MPS) with a finite bond dimension in the thermodynamic limit -- an independent and explicit construction for the BDI class is given in a concurrent work (Jones, Bibo, Jobst, Pollmann, Smith, Verresen, arXiv:2105.12143); (4) for BDI models with even integer topological invariant, all non-zero eigenvalues of the transfer matrix are identified as products of zeros and inverse zeros of the aforementioned polynomial. We show how general models in these classes can be obtained by taking limits of the models studied in this work, which can be used to apply limits of our results to the general case. To the best of our knowledge, these results constitute the first application of Days formula and Gorodetskys formula from the theory of Toeplitz determinants to the context of many-body quantum physics.
We consider the realization of universal quantum computation through braiding of Majorana fermions supplemented by unprotected preparation of noisy ancillae. It has been shown by Bravyi [Phys. Rev. A 73, 042313 (2006)] that under the assumption of perfect braiding operations, universal quantum computation is possible if the noise rate on a particular 4-fermion ancilla is below 40%. We show that beyond a noise rate of 89% on this ancilla the quantum computation can be efficiently simulated classically: we explicitly show that the noisy ancilla is a convex mixture of Gaussian fermionic states in this region, while for noise rates below 53% we prove that the state is not a mixture of Gaussian states. These results were obtained by generalizing concepts in entanglement theory to the setting of Gaussian states and their convex mixtures. In particular we develop a complete set of criteria, namely the existence of a Gaussian-symmetric extension, which determine whether a state is a convex mixture of Gaussian states.
Certain physical systems that one might consider for fault-tolerant quantum computing where qubits do not readily interact, for instance photons, are better suited for measurement-based quantum-computational protocols. Here we propose a measurement-based model for universal quantum computation that simulates the braiding and fusion of Majorana modes. To derive our model we develop a general framework that maps any scheme of fault-tolerant quantum computation with stabilizer codes into the measurement-based picture. As such, our framework gives an explicit way of producing fault-tolerant models of universal quantum computation with linear optics using protocols developed using the stabilizer formalism. Given the remarkable fault-tolerant properties that Majorana modes promise, the main example we present offers a robust and resource efficient proposal for photonic quantum computation.
We introduce a new method for representing the low energy subspace of a bosonic field theory on the qubit space of digital quantum computers. This discretization leads to an exponentially precise description of the subspace of the continuous theory thanks to the Nyquist-Shannon sampling theorem. The method makes the implementation of quantum algorithms for purely bosonic systems as well as fermion-boson interacting systems feasible. We present algorithmic circuits for computing the time evolution of these systems. The complexity of the algorithms scales polynomially with the system size. The algorithm is a natural extension of the existing quantum algorithms for simulating fermion systems in quantum chemistry and condensed matter physics to systems involving bosons and fermion-boson interactions and has a broad variety of potential applications in particle physics, condensed matter, etc. Due to the relatively small amount of additional resources required by the inclusion of bosons in our algorithm, the simulation of electron-phonon and similar systems can be placed in the same near-future reach as the simulation of interacting electron systems. We benchmark our algorithm by implementing it for a $2$-site Holstein polaron problem on an Atos Quantum Learning Machine (QLM) quantum simulator. The polaron quantum simulations are in excellent agreement with the results obtained by exact diagonalization.
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