The diamond distance can measure the error between real and ideal quantum processes. We can calculate the distance for trace-preserving channels using convex optimization. However, convex optimization cannot solve the distance for non-trace-preserving channels because they are nonlinear. We generalize the diamond distance for non-trace-preserving channels. We first compute the distance for a simple measurement and post-selection channel, then generalize the method to any non-trace-preserving channels. As an example, we analyze the Knill-Laflamme-Milburn (KLM) protocol in linear optical quantum computing with physical imperfections. Our method provides a well-behaved measure for general quantum channels, which can bridge the physical imperfections with the logical errors in quantum circuits. Hence, it has broad applications in analyzing real systems in quantum information processing, such as quantum computers, quantum teleportation, measurement-based quantum computation, etc.
The diamond distance can measure the error between real and ideal quantum processes. We can calculate the distance for trace-preserving channels using convex optimization. However, convex optimization cannot solve the distance for non-trace-preserving channels because they are nonlinear. We generalize the diamond distance for non-trace-preserving channels. We first compute the distance for a simple measurement and post-selection channel, then generalize the method to any non-trace-preserving channels. As an example, we analyze the Knill-Laflamme-Milburn (KLM) protocol in linear optical quantum computing with physical imperfections. Our method provides a well-behaved measure for general quantum channels, which can bridge the physical imperfections with the logical errors in quantum circuits. Hence, it has broad applications in analyzing real systems in quantum information processing, such as quantum computers, quantum teleportation, measurement-based quantum computation, etc.
We study the problem of measuring errors in non-trace-preserving quantum operations, with a focus on their impact on quantum computing. We propose an error metric that efficiently provides an upper bound on the trace distance between the normalized output states from imperfect and ideal operations, while remaining compatible with the diamond distance. As a demonstration of its application, we apply our metric in the analysis of a lossy beam splitter and a nondeterministic conditional sign-flip gate, two primary non-trace-preserving operations in the Knill-Laflamme-Milburn protocol. We then turn to the leakage errors of neutral-atom quantum computers, finding that these errors scale worse than previously anticipated, implying a more stringent fault-tolerant threshold. We also assess the quantum Zeno gate's error using our metric. In a broader context, we discuss the potential of our metric to analyze general postselected protocols, where it can be employed to study error propagation and estimate thresholds in fault-tolerant quantum computing. The results highlight the critical role of our proposed error metric in understanding and addressing challenges in practical quantum information processing.
Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels — that is, not completely positive — can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simulating unphysical dynamics onto the task of simulating linear combinations of quantum states. Our second measure benchmarks the quantitative advantages that a non-completely-positive map can provide in discrimination-based quantum games. Notably, we show that for any trace-preserving map, the quantities both reduce to a fundamental distance measure: the diamond norm, thus endowing this norm with new operational meanings in the characterisation of linear maps. We discuss applications of our results to structural physical approximations of positive maps, quantification of non-Markovianity, and bounding the cost of error mitigation.
Quantum process tomography, the task of estimating an unknown quantum channel, is a central problem in quantum information theory. A long-standing open question is how many uses of an unknown channel are required to learn it in diamond distance, the standard metric for distinguishing quantum processes. While quantum state tomography is well understood, for general channels the problem remained open beyond the unitary case. Here we establish the query complexity of channel tomography with optimal dependence on the dimension parameters, at any fixed constant accuracy. We design an algorithm showing that any channel with input/output dimensions din,dout and Kraus rank at most k can be learned to accuracy ε using O(dindoutk/ε2) channel uses. Conversely, we prove that Ω(dindoutk) uses are necessary at constant accuracy and that, for non-minimal Kraus rank, a separate Ω(1/ε2) contribution is unavoidable. Since channels subsume states, unitaries, isometries, and measurements as special cases, our protocol provides a unified framework for these tomography tasks, yielding new guarantees for isometry and measurement tomography while recovering known optimal scalings for state and unitary tomography. Our algorithm follows the natural strategy of performing optimal tomography on the Choi state. The main technical contribution is to show that this suffices to control the induced diamond-distance error, avoiding the dimension loss incurred by a naive conversion from Choi-state trace distance to channel diamond distance. The protocol uses the channel non-adaptively to prepare Choi-state copies, purifies them in parallel, and performs optimal pure-state tomography on the resulting purifications. Hence, we reduce channel tomography to pure-state tomography.
The framework of postselection is becoming more and more important in various recent directions in Quantum Computation research. Postselection renders simple computational models able to perform general quantum computation. This was first observed for the linear optics model [E. Knill, R. Laflamme, G. J. Milburn, Nature 409, 46 (2001)], and has since provided us with many near-term candidates for the quantum advantage, commuting computations [M. J. Bremner, R. Jozsa, D. J. Shepherd, Proc. R. Soc. A 467, 459 (2011)] being the first. To facilitate the discussion of errors in the presence of postselection, we define and characterize trace-induced distance and diamond distance of postselected computations. We show counterexamples to simple properties that one would expect of any distance measure; the properties of convexity (when considering only the pure-state inputs would suffice), contractivity, and subadditivity of errors. On the positive side, we prove that certain weaker versions of contractivity and subadditivity and a number of other properties are preserved in the postselected setting. We achieve this via a "conversion lemma" that translates any inequality from the standard to the postselected setting.
In this work, we present an in-depth study of average-case quantum distances introduced in Maciejewski et al. (2022). The average-case distances approximate, up to the relative error, the average Total-Variation (TV) distance between measurement outputs of two quantum processes, in which quantum objects of interest (states, measurements, or channels) are intertwined with random quantum circuits. Contrary to conventional distances, such as trace distance or diamond norm, they quantifyaverage-casestatistical distinguishability via random quantum circuits. We prove that once a family of random circuits forms an$\delta −approximate4−design,with\delta =o(d^{-8}),thentheaverage−casedistancescanbeapproximatedbysimpleexplicitfunctionsthatcanbeexpressedviasimpledegreetwopolynomialsinobjectsofinterest.Forsystemsofmoderatedimension,theycanbeeasilyexplicitlycomputed–nooptimizationisneededasopposedtodiamondnormdistancebetweenchannelsoroperationaldistancebetweenmeasurements.Weprovethatthosefunctions,whichwecallquantumaverage−casedistances,haveaplethoraofdesirableproperties,suchassubadditivityw.r.t.tensorproducts,jointconvexity,and(restricted)data−processinginequalities.Notably,alldistancesutilizetheHilbert−Schmidt(HS)norm,whichprovidesthisnormwithanewoperationalinterpretation.Wealsoprovideupperboundsonthemaximalratiobetweenworst−caseandaverage−casedistances,andforeachofthem,weprovideanexamplethatsaturatesthebound.Specifically,weshowthatforeachdimensiondthisratioisatmostd^{\frac {1}{2}}, d, d^{\frac {3}{2}}$for states, measurements, and channels, respectively. To support the practical usefulness of our findings, we study multiple examples in which average-case quantum distances can be calculated analytically.
The diamond distance between two quantum channels determines the optimal probability of distinguishing them from a single use. We give a quantum algorithm that estimates the diamond distance between two unknown d-dimensional unitary channels to within additive error ε, using O(d/ε) queries to the unitary channels and their inverses. The query complexity is optimal, as it matches the lower bound due to Jeon and Oh (npj Quantum Inf. 2026).