Efficient Measurement of Quantum Dynamics via Compressive Sensing

Abstract

The resources required to characterize the dynamics of engineered quantum systems-such as quantum computers and quantum sensors-grow exponentially with system size. Here we adapt techniques from compressive sensing to exponentially reduce the experimental configurations required for quantum process tomography. Our method is applicable to processes that are nearly sparse in a certain basis and can be implemented using only single-body preparations and measurements. We perform efficient, highfidelity estimation of process matrices of a photonic two-qubit logic gate. The database is obtained under various decoherence strengths. Our technique is both accurate and noise robust, thus removing a key roadblock to the development and scaling of quantum technologies. Understanding and controlling the world at the nanoscale-be it in biological, chemical or physical phenomena-requires quantum mechanics. It is therefore essential to characterize and monitor realistic complex quantum systems that inevitably interact with typically uncontrollable environments. One of the most general descriptions of the dynamics of an open quantum system is a quantum map-typically represented by a process matrix [1]. Methods to identify this matrix are collectively known as quantum process tomography (QPT) [1,2]. For a d-dimensional quantum system, they require Oðd 4 Þ experimental configurations: combinations of input states, on which the process acts, and a set of output observables. For a system of n qubits-two level quantum systemsd ¼ 2 n . The required physical resources hence scale exponentially with system size. Recently, a number of alternative methods have been developed for efficient and selective estimation of quantum processes [3]. However, full characterization of quantum dynamics of comparably small systems, such as an 8-qubit ion trap [4], would still require over a billion experimental configurations, clearly impractical. So far, process tomography has therefore been limited by experimental and off-line computational resources, to systems of 2 and 3 qubits [5][6][7].Here we adapt techniques from compressive sensing to develop an experimentally efficient method for QPT. It requires only Oðs logdÞ configurations if the process matrix is s compressible in some known basis, i.e., it is nearly sparse in that it can be well approximated by an s-sparse process matrix. This is usually the case, because engineered quantum systems aim to implement a unitary process which is maximally sparse in its eigenbasis. In practice, as observed in liquid-state NMR [8], photonics [5,9,10], ion traps [11], and superconducting circuits [6], a near-unitary process will still be nearly sparse in this basis, and still compressible. The near sparsity is due to few dominant system environment interactions. This is more apparent for weakly decohering systems [12].We experimentally demonstrate our algorithm by estimating the 240 real parameters of the process matrix of a canonical photonic two-qubit gate, Fig. 1, from a reduced number of configurations. From...

Application of compressed sensing to the simulation of atomic systems

“…3, we select the one that has the largest number of zero coefficients: the sparsest solution. This procedure turns out to be equivalent to the so-called basis pursuit (BP) optimization problem (18) min g jgj 1 subject to Fg ¼ h; [5] which is what one solves in practice (where…”

“…This extra information can be used by the CS method to significantly reduce the number of measurements needed to reconstruct a signal. CS has been successfully applied to data acquisition in many different areas (3), including the improvement of the resolution of medical magnetic resonance imaging (4) and the experimental study of atomic and quantum systems (5)(6)(7).…”

Direct Fidelity Estimation from Few Pauli Measurements

“…One may also compare our method with recent proposals for tomography for restricted classes of quantum states [24][25][26][27]. Another important direction is to find better techniques for sampling the importanceweighting distribution PrðkÞ for different classes of states.…”

Continuous-variable quantum process tomography with squeezed-state probes

“…In some cases, scalable quantum process reconstruction may be achieved e.g. by approximating the operator χ by a matrix product state [22,23] or by using compressed sensing techniques [24][25][26].…”