One of the main aims in the field of quantum simulation is to achieve a quantum speedup, often referred to as "quantum computational supremacy," referring to the experimental realization of a quantum device that computationally outperforms classical computers. In this work, we show that one can devise versatile and feasible schemes of two-dimensional, dynamical, quantum simulators showing such a quantum speedup, building on intermediate problems involving nonadaptive, measurement-based, quantum computation. In each of the schemes, an initial product state is prepared, potentially involving an element of randomness as in disordered models, followed by a short-time evolution under a basic translationally invariant Hamiltonian with simple nearest-neighbor interactions and a mere sampling measurement in a fixed basis. The correctness of the final-state preparation in each scheme is fully efficiently certifiable. We discuss experimental necessities and possible physical architectures, inspired by platforms of cold atoms in optical lattices and a number of others, as well as specific assumptions that enter the complexity-theoretic arguments. This work shows that benchmark settings exhibiting a quantum speedup may require little control, in contrast to universal quantum computing. Thus, our proposal puts a convincing experimental demonstration of a quantum speedup within reach in the near term.

A central question in quantum computation is to identify the resources that are responsible for quantum speed-up. Quantum contextuality has been recently shown to be a resource for quantum computation with magic states for odd-prime dimensional qudits and two-dimensional systems with real wave functions. The phenomenon of state-independent contextuality poses a priori an obstruction to characterizing the case of regular qubits, the fundamental building block of quantum computation. Here, we establish contextuality of magic states as a necessary resource for a large class of quantum computation schemes on qubits. We illustrate our result with a concrete scheme related to measurement-based quantum computation. DOI: 10.1103/PhysRevLett.119.120505 The model of quantum computation by state injection (QCSI) [1] is a leading paradigm of fault-tolerance quantum computation. Therein, quantum gates are restricted to belong to a small set of classically simulable gates, called Clifford gates [2], that admit simple fault-tolerant implementations [3]. Universal quantum computation is achieved via injection of magic states [1], which are the source of quantum computational power of the model.A central question in QCSI is to characterize the physical properties that magic states need to exhibit in order to serve as universal resources. In this regard, quantum contextuality has recently been established as a necessary resource for QCSI. This was first achieved for quopit systems [4,5], where the local Hilbert space dimension is an odd prime power, and subsequently for local dimension two with the case of rebits [6]. In the latter, the density matrix is constrained to be real at all times.In this Letter we ask "Can contextuality be established as a computational resource for QCSI on qubits?" This is not a straightforward extension of the quopit case because the multiqubit setting is complicated by the presence of stateindependent contextuality among Pauli observables [7,8]. Consequently, every quantum state of n ≥ 2 qubits is contextual with respect to Pauli measurements, including the completely mixed one [5]. It is thus clear that contextuality of magic states alone cannot be a computational resource for every QCSI scheme on qubits.Yet, there exist qubit QCSI schemes for which contextuality of magic states is a resource, and we identify them in this Letter. Specifically, we consider qubit QCSI schemes M O that satisfy the following two constraints: (C1) Resource character. There exists a quantum state that does not exhibit contextuality with respect to measurements available in M O . (C2) Tomographic completeness. For any state ρ, the expectation value of any Pauli observable can be inferred via the allowed operations of the scheme.The motivation for these constraints is the following. Condition (C1) constitutes a minimal principle that unifies, simplifies and extends the quopit [5] and rebit [6] settings. While seemingly a weak constraint, it excludes the possibility of Mermin-type state-independent contextuality [7,8] amo...

Quantum normalizer circuits were recently introduced as generalizations of Clifford circuits [1]: a normalizer circuit over a finite Abelian group G is composed of the quantum Fourier transform (QFT) over G, together with gates which compute quadratic functions and automorphisms. In [1] it was shown that every normalizer circuit can be simulated efficiently classically. This result provides a nontrivial example of a family of quantum circuits that cannot yield exponential speed-ups in spite of usage of the QFT, the latter being a central quantum algorithmic primitive. Here we extend the aforementioned result in several ways. Most importantly, we show that normalizer circuits supplemented with intermediate measurements can also be simulated efficiently classically, even when the computation proceeds adaptively. This yields a generalization of the Gottesman-Knill theorem (valid for n-qubit Clifford operations [2, 3]) to quantum circuits described by arbitrary finite Abelian groups. Moreover, our simulations are twofold: we present efficient classical algorithms to sample the measurement probability distribution of any adaptive-normalizer computation, as well as to compute the amplitudes of the state vector in every step of it. Finally we develop a generalization of the stabilizer formalism [2, 3] relative to arbitrary finite Abelian groups: for example we characterize how to update stabilizers under generalized Pauli measurements and provide a normal form of the amplitudes of generalized stabilizer states using quadratic functions and subgroup cosets.

We describe schemes of quantum computation with magic states on qubits for which contextuality and negativity of the Wigner function are necessary resources possessed by the magic states. These schemes satisfy a constraint. Namely, the non-negativity of Wigner functions must be preserved under all available measurement operations. Furthermore, we identify stringent consistency conditions on such computational schemes, revealing the general structure by which negativity of Wigner functions, hardness of classical simulation of the computation, and contextuality are connected.