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We consider a model of quantum computation using qubits where it is possible to measure whether a given pair are in a singlet (total spin $0$) or triplet (total spin $1$) state. The physical motivation is that we can do these measurements in a way that is protected against revealing other information so long as all terms in the Hamiltonian are $SU(2)$-invariant. We conjecture that this model is equivalent to BQP. Towards this goal, we show: (1) this model is capable of universal quantum computation with polylogarithmic overhead if it is supplemented by single qubit $X$ and $Z$ gates. (2) Without any additional gates, it is at least as powerful as the weak model of permutational quantum computation of Jordan[1, 2]. (3) With postselection, the model is equivalent to PostBQP.
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The main obstacles to the realization of high-fidelity quantum gates are the control errors arising from inaccurate manipulation of a quantum system and the decoherence caused by the interaction between the quantum system and its environment. Nonadiabatic holonomic quantum computation allows for high-speed implementation of whole-geometric quantum gates, making quantum computation robust against control errors. Dynamical decoupling provides an effective method to protect quantum gates against environment-induced decoherence, regardless of collective decoherence or independent decoherence. In this paper, we put forward a protocol of nonadiabatic holonomic quantum computation protected by dynamical decoupling . Due to the combination of nonadiabatic holonomic quantum computation and dynamical decoupling, our protocol not only possesses the intrinsic robustness against control errors but also protects quantum gates against environment-induced decoherence.
Quantum adiabatic evolution, an important fundamental concept inphysics, describes the dynamical evolution arbitrarily close to the instantaneous eigenstate of a slowly driven Hamiltonian. In most systems undergoing spontaneous symmetry-breaking transitions, their two lowest eigenstates change from non-degenerate to degenerate. Therefore, due to the corresponding energy-gap vanishes, the conventional adiabatic condition becomes invalid. Here we explore the existence of quantum adiabatic evolutions in spontaneous symmetry-breaking transitions and derive a symmetry-dependent adiabatic condition. Because the driven Hamiltonian conserves the symmetry in the whole process, the transition between different instantaneous eigenstates with different symmetries is forbidden. Therefore, even if the minimum energy-gap vanishes, symmetry-protected quantum adiabatic evolutioncan still appear when the driven system varies according to the symmetry-dependent adiabatic condition. This study not only advances our understandings of quantum adiabatic evolution and spontaneous symmetry-breaking transitions, but also provides extensive applications ranging from quantum state engineering, topological Thouless pumping to quantum computing.
A preliminary overview of measurement-based quantum computation in the setting of symmetry and topological phases of quantum matter is given. The underlying mechanism for universal quantum computation by teleportation or symmetry are analyzed, with the emphasis on the relation with tensor-network states in the presence of various symmetries. Perspectives are also given for the role of symmetry and phases of quantum matter in measurement-based quantum computation and fault tolerance.
Buhrman, Cleve and Wigderson (STOC98) showed that for every Boolean function f : {-1,1}^n to {-1,1} and G in {AND_2, XOR_2}, the bounded-error quantum communication complexity of the composed function f o G equals O(Q(f) log n), where Q(f) denotes the bounded-error quantum query complexity of f. This is in contrast with the classical setting, where it is easy to show that R^{cc}(f o G) < 2 R(f), where R^{cc} and R denote bounded-error communication and query complexity, respectively. Chakraborty et al. (CCC20) exhibited a total function for which the log n overhead in the BCW simulation is required. We improve upon their result in several ways. We show that the log n overhead is not required when f is symmetric, generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing05). This upper bound assumes a shared entangled state, though for most symmetric functions the assumed number of entangled qubits is less than the communication and hence could be part of the communication. To prove this, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function. In view of our first result, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive. We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2. We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model, even if the parties are allowed to share an arbitrary prior entangled state for free.
Periodically driven quantum systems, known as Floquet systems, have been a focus of non-equilibrium physics in recent years, thanks to their rich dynamics. Not only time-periodic systems exhibit symmetries similar to those in spatially periodic systems, but they also display novel behavior due to symmetry breaking. Characterizing such dynamical symmetries is crucial, but the task is often challenging, due to limited driving strength and the lack of an experimentally accessible characterization protocol. Here, we show how to characterize dynamical symmetries including parity, rotation, and particle-hole symmetry by observing the symmetry-induced selection rules between Floquet states. Specifically, we exploit modulated quantum driving to reach the strong light-matter coupling regime and we introduce a protocol to experimentally extract the transition elements between Floquet states from the coherent evolution of the system. Using the nitrogen-vacancy center in diamond as an experimental testbed, we apply our methods to observe symmetry-protected dark states and dark bands, and the coherent destruction of tunneling effect. Our work shows how to exploit the quantum control toolkit to study dynamical symmetries that can arise in topological phases of strongly-driven Floquet systems.
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