We develop a method for quantum process tomography that combines the efficiency of compressed sensing with the robustness of randomized benchmarking. Our method is robust to state preparation and measurement errors, and it achieves a quadratic speedup over conventional tomography when the unknown process is a generic unitary evolution. Our method is based on PhaseLift, a convex programming technique for phase retrieval. We show that this method achieves approximate recovery of almost all signals, using measurements sampled from spherical or unitary 2-designs. This is the first positive result on PhaseLift using 2-designs. We also show that exact recovery of all signals is possible using unitary 4-designs. Previous positive results for PhaseLift required spherical 4-designs, while PhaseLift was known to fail in certain cases when using spherical 2-designs.
We develop a method for quantum process tomography that combines the efficiency of compressed sensing with the robustness of randomized benchmarking. Our method is robust to state preparation and measurement errors, and it achieves a quadratic speedup over conventional tomography when the unknown process is a generic unitary evolution. Our method is based on PhaseLift, a convex programming technique for phase retrieval. We show that this method achieves approximate recovery of almost all signals, using measurements sampled from spherical or unitary 2-designs. This is the first positive result on PhaseLift using 2-designs. We also show that exact recovery of all signals is possible using unitary 4-designs. Previous positive results for PhaseLift required spherical 4-designs, while PhaseLift was known to fail in certain cases when using spherical 2-designs.
We consider a variant of the phase retrieval problem, where vectors are replaced by unitary matrices, i.e., the unknown signal is a unitary matrix U, and the measurements consist of squared inner products |tr(C†U)|2with unitary matrices C that are chosen by the observer. This problem has applications to quantum process tomography, when the unknown process is a unitary operation. We show that PhaseLift, a convex programming algorithm for phase retrieval, can be adapted to this matrix setting, using measurements that are sampled from unitary 4- and 2-designs. In the case of unitary 4-design measurements, we show that PhaseLift can reconstruct all unitary matrices, using a nearoptimal number of measurements. This extends previous work on PhaseLift using spherical 4-designs. In the case of unitary 2-design measurements, we show that PhaseLift still works pretty well on average: it recovers almost all signals, up to a constant additive error, using a near-optimal number of measurements. These 2-design measurements are convenient for quantum process tomography, as they can be implemented via randomized benchmarking techniques. This is the first positive result on PhaseLift using 2-designs.
We study quantum process tomography given the prior information that the map is a unitary or close to a unitary process. We show that a unitary map on a d-level system is completely characterized by a minimal set of d2+d elements associated with a collection of POVMs, in contrast to the ${d}^{4}\ensuremath{-}{d}^{2}$ elements required for a general completely positive trace-preserving map. To achieve this lower bound, one must probe the map with particular sets of d pure states. We further compare the performance of different estimators inspired by compressed sensing, to reconstruct a near-unitary process from such data. We find that when we have accurate prior information, an appropriate compressed sensing method reduces the required data needed for high-fidelity estimation, and different estimators applied to the same data are sensitive to different types of noise. Convex optimization inspired by the techniques of compressed sensing can therefore be used both as indicators of error models and to validate the use of the prior assumptions.
Quantum process tomography is the task of reconstructing unknown quantum channels from measured data. In this work, we introduce compressed sensing-based methods that facilitate the reconstruction of quantum channels of low Kraus rank. Our main contribution is the analysis of a natural measurement model for this task: We assume that data is obtained by sending pure states into the channel and measuring expectation values on the output. Neither ancillary systems nor coherent operations across multiple channel uses are required. Most previous results on compressed process reconstruction reduce the problem to quantum state tomography on the channel's Choi matrix. While this ansatz yields recovery guarantees from an essentially minimal number of measurements, physical implementations of such schemes would typically involve ancillary systems. A priori, it is unclear whether a measurement model tailored directly to quantum process tomography might require more measurements. We establish that this is not the case.Technically, we prove recovery guarantees for three different reconstruction algorithms. The reconstructions are based on a trace, diamond, and ℓ2 -norm minimization, respectively. Our recovery guarantees are uniform in the sense that with one random choice of measurement settings all quantum channels can be recovered equally well. Moreover, stability against arbitrary measurement noise and robustness against violations of the low-rank assumption is guaranteed. Numerical studies demonstrate the feasibility of the approach.
Characterizing quantum processes is a key task in the development of quantum technologies, especially at the noisy intermediate scale of today's devices. One method for characterizing processes is randomized benchmarking, which is robust against state preparation and measurement errors and can be used to benchmark Clifford gates. Compressed sensing techniques achieve full tomography of quantum channels essentially at optimal resource efficiency. In this Letter, we show that the favorable features of both approaches can be combined. For characterizing multiqubit unitary gates, we provide a rigorously guaranteed and practical reconstruction method that works with an essentially optimal number of average gate fidelities measured with respect to random Clifford unitaries. Moreover, for general unital quantum channels, we provide an explicit expansion into a unitary 2-design, allowing for a practical and guaranteed reconstruction also in that case. As a side result, we obtain a new statistical interpretation of the unitarity-a figure of merit characterizing the coherence of a process.
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, high-fidelity 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.
In the light of the progress in quantum technologies, the task of verifying the correct functioning of processes and obtaining accurate tomographic information about quantum states becomes increasingly important. Compressed sensing, a machinery derived from the theory of signal processing, has emerged as a feasible tool to perform robust and significantly more resource-economical quantum state tomography for intermediate-sized quantum systems. In this work, we provide a comprehensive analysis of compressed sensing tomography in the regime in which tomographically complete data is available with reliable statistics from experimental observations of a multi-mode photonic architecture. Due to the fact that the data is known with high statistical significance, we are in a position to systematically explore the quality of reconstruction depending on the number of employed measurement settings, randomly selected from the complete set of data, and on different model assumptions. We present and test a complete prescription to perform efficient compressed sensing and are able to reliably use notions of model selection and cross validation to account for experimental imperfections and finite counting statistics. Thus, we establish compressed sensing as an effective tool for quantum state tomography, specifically suited for photonic systems.