Tomography of Quantum States in Small Dimensions

Sizhuo Yu, Jean‐Michel Gillet

Kai Li, Le Yang, Dai HongYi +1 more

We explore the impact of quantum algorithms on classical-data problems if quantum algorithms can be implemented on quantum computers. It is underlined in this paper that both state estimator design problems for classical linear systems and quantum state tomography can be considered as classical-data problems. Our recent research indicated that the time complexity of state estimator design problems can be reduced to O(qn) in comparison to the time complexity O(n6) in traditional classical algorithms when quantum states can be efficiently prepared, the system matrix is sparse, and both the condition number κ and the reciprocal of precision ε are small in size O(poly log(n)), where n is the dimension of the state x(t) and q is the dimension of the input u(t). This is in contrast to the further observation that the time complexity of state estimator design problems is O(n2) when quantum states cannot be efficiently prepared. The time complexity of quantum state tomography with dimension d is O(d4) when classical algorithms are adopted. We have demonstrated that if the system matrix is sparse, and both the condition number κ and the reciprocal of precision ε are small in size O(poly log(d)), quantum algorithms can reduce the time complexity of quantum state tomography to O(dpoly log d) or O(d3) when quantum states can or cannot be efficiently prepared. In other words, preparing quantum states efficiently has become a bottleneck constraint for quantum acceleration. Our recent researches can be regarded as a novel attempt of enriching application area of quantum computation.

Jeff S. Lundeen, Alejandro Feito, Hendrik B. Coldenstrodt-Ronge +6 more

Alireza Shabani, Robert L. Kosut, HERSCHEL A. RABITZ

The characterization of a decoherence process is among the central challenges in quantum physics. A major difficulty with current quantum process tomography methods is the enormous number of experiments needed to accomplish a tomography task. Here we present a highly efficient method for tomography of a quantum process that has a small number of significant elements. Our method is based on the compressed sensing techniques being used in information theory. In this new method, for a system with Hilbert space dimension n and a process matrix of dimension n × n with sparsity s, the required number of experimental configurations is O(s log n). This heralds a logarithmic advantage in contrast to other methods of quantum process tomography. More specifically, for q-qubits with n = 2, the scaling of resources is O(sq) — linear in the product of sparsity and number of qubits.

Cristina Butucea, Mădălin Guţǎ, Theodore Kypraios

The estimation of high dimensional quantum states is an important statistical problem arising in current quantum technology applications. A key example is the tomography of multiple ions states, employed in the validation of state preparation in ion trap experiments (Häffner et al 2005 Nature 438 643 ). Since full tomography becomes unfeasible even for a small number of ions, there is a need to investigate lower dimensional statistical models which capture prior information about the state, and to devise estimation methods tailored to such models. In this paper we propose several new methods aimed at the efficient estimation of low rank states and analyse their performance for multiple ions tomography. All methods consist in first computing the least squares estimator, followed by its truncation to an appropriately chosen smaller rank. The latter is done by setting eigenvalues below a certain 'noise level' to zero, while keeping the rest unchanged, or normalizing them appropriately. We show that (up to logarithmic factors in the space dimension) the mean square error of the resulting estimators scales as where r is the rank, is the dimension of the Hilbert space, and N is the number of quantum samples. Furthermore we establish a lower bound for the asymptotic minimax risk which shows that the above scaling is optimal. The performance of the estimators is analysed in an extensive simulations study, with emphasis on the dependence on the state rank, and the number of measurement repetitions. We find that all estimators perform significantly better than the least squares, with the 'physical estimator' (which is a bona fide density matrix) slightly outperforming the other estimators.

B. P. Lanyon, Christine Maier, Milan Holzäpfel +10 more