Mitigating Quantum Gate Errors for Variational Eigensolvers Using Hardware-Inspired Zero-Noise Extrapolation

Variational quantum algorithms have emerged as a cornerstone of contemporary quantum algorithms research. Practical implementations of these algorithms, despite offering certain levels of robustness against systematic errors, show a decline in performance due to the presence of stochastic errors and limited coherence time. In this work, we develop a recipe for mitigating quantum gate errors for variational algorithms using zero-noise extrapolation. We introduce an experimentally amenable method to control error strength in the circuit. We utilise the fact that gate errors in a physical quantum device are distributed inhomogeneously over different qubits and pairs thereof. As a result, one can achieve different circuit error sums based on the manner in which abstract qubits in the circuit are mapped to a physical device. We find that the estimated energy in the variational approach is approximately linear with respect to the circuit error sum (CES). Consequently, a linear fit through the energy-CES data, when extrapolated to zero CES, can approximate the energy estimated by a noiseless variational algorithm. We demonstrate this numerically and further prove that the approximation is exact if the two-qubit gates in the circuits are arranged in the form of a regular graph.Comment: 9 pages, 2 figure

Ion native variational ansatz for quantum approximate optimization

Variational quantum algorithms involve training parameterized quantum circuits using a classical co-processor. An important variational algorithm, designed for combinatorial optimization, is the quantum approximate optimization algorithm. Realization of this algorithm on any modern quantum processor requires either embedding a problem instance into a Hamiltonian or emulating the corresponding propagator by a gate sequence. For a vast range of problem instances this is impossible due to current circuit depth and hardware limitations. Hence we adapt the variational approach -- using ion native Hamiltonians -- to create ansatze families that can prepare the ground states of more general problem Hamiltonians. We analytically determine symmetry protected classes that make certain problem instances inaccessible unless this symmetry is broken. We exhaustively search over six qubits and consider upto twenty circuit layers, demonstrating that symmetry can be broken to solve all problem instances of the Sherrington-Kirkpatrick Hamiltonian. Going further, we numerically demonstrate training convergence and level-wise improvement for up to twenty qubits. Specifically these findings widen the class problem instances which might be solved by ion based quantum processors. Generally these results serve as a test-bed for quantum approximate optimization approaches based on system native Hamiltonians and symmetry protection.Comment: 9 pages; 5 figures; REVTe

Bell-CHSH non-locality and entanglement from a unified framework

Non-classical probability is a defining feature of quantum mechanics. This paper develops a formalism that exhibits explicitly, the manner in which rules of classical probability break down in the quantum domain. Thereby, a framework is set up which allows for construction of signatures for non-classicality of states in a systematic manner. Using this, conditions for non-locality and entanglement are shown to emerge from a break down of classical probability rules. Bell-CHSH non-locality is derived for any bipartite systems and entanglement inequalities are obtained for coupled two level systems only

Revisiting integer factorization using closed timelike curves

Closed timelike curves are relativistically valid objects allowing time travel to the past. Treating them as computational objects opens the door to a wide range of results which cannot be achieved using non-relativistic quantum mechanics. Recently, research in classical and quantum computation has focused on effectively harnessing the power of these curves. In particular, Brun (Found Phys Lett 16:245-253, 2003) has shown that CTCs can be utilized to efficiently solve problems like factoring and quantified satisfiability problem. In this paper, we find a flaw in Brun's algorithm and propose a modified algorithm to circumvent the flaw

Tensor networks in machine learning

A tensor network is a type of decomposition used to express and approximate large arrays of data. A given data-set, quantum state or higher dimensional multi-linear map is factored and approximated by a composition of smaller multi-linear maps. This is reminiscent to how a Boolean function might be decomposed into a gate array: this represents a special case of tensor decomposition, in which the tensor entries are replaced by 0, 1 and the factorisation becomes exact. The collection of associated techniques are called, tensor network methods: the subject developed independently in several distinct fields of study, which have more recently become interrelated through the language of tensor networks. The tantamount questions in the field relate to expressability of tensor networks and the reduction of computational overheads. A merger of tensor networks with machine learning is natural. On the one hand, machine learning can aid in determining a factorization of a tensor network approximating a data set. On the other hand, a given tensor network structure can be viewed as a machine learning model. Herein the tensor network parameters are adjusted to learn or classify a data-set. In this survey we recover the basics of tensor networks and explain the ongoing effort to develop the theory of tensor networks in machine learning.Comment: 7 page