Solomonoff's universal inductive inference [Inf. Control. 1964] is one of the most general frameworks for learning from sequential data and predicting future observations. It provides prediction guarantees for any computable stochastic source. We ask whether an analogous universal theory of induction can be developed for quantum systems. To this end, we introduce universal quantum inductive inference, a framework for learning and predicting quantum sources that may exhibit arbitrary correlations and entanglement across time. Given the outcomes of past measurements and the corresponding post-measurement quantum systems, the learner produces a joint state that predicts the next measurement outcome and post-measurement system while preserving correlations with the past systems. We model a quantum source as a multipartite quantum state whose description is generated by an unknown $s$ -bit program within a given time bound. The learner is required to produce a state that is $ε$ -close in trace distance to the target with probability at least $1-δ$ . We establish an information-theoretic inference algorithm with round complexity $O(sε^{-2}δ^{-1})$ . We further prove a matching lower bound in the relevant parameter regime, even for classical sources. As a second result, we construct a non-i.i.d. state tomography algorithm that outputs a classical description of the conditional state of the next system given past measurement outcomes. Our information-theoretic tomography algorithm achieves round complexity $O(s\min\{2^s,2^n\}ε^{-2}δ^{-1})$ , where $n$ is the number of qubits received in each round. Finally, we investigate the computational hardness of universal quantum inductive inference under quantum cryptographic assumptions.Solomonoff's universal inductive inference [Inf. Control. 1964] is one of the most general frameworks for learning from sequential data and predicting future observations. It provides prediction guarantees for any computable stochastic source. We ask whether an analogous universal theory of induction can be developed for quantum systems. To this end, we introduce universal quantum inductive inference, a framework for learning and predicting quantum sources that may exhibit arbitrary correlations and entanglement across time. Given the outcomes of past measurements and the corresponding post-measurement quantum systems, the learner produces a joint state that predicts the next measurement outcome and post-measurement system while preserving correlations with the past systems. We model a quantum source as a multipartite quantum state whose description is generated by an unknown $s$-bit program within a given time bound. The learner is required to produce a state that is $ε$-close in trace distance to the target with probability at least $1-δ$. We establish an information-theoretic inference algorithm with round complexity $O(sε^{-2}δ^{-1})$. We further prove a matching lower bound in the relevant parameter regime, even for classical sources. As a second result, we construct a non-i.i.d. state tomography algorithm that outputs a classical description of the conditional state of the next system given past measurement outcomes. Our information-theoretic tomography algorithm achieves round complexity $O(s\min\{2^s,2^n\}ε^{-2}δ^{-1})$, where $n$ is the number of qubits received in each round. Finally, we investigate the computational hardness of universal quantum inductive inference under quantum cryptographic assumptions.

quant-pharXiv:2609.36912Sep 30, 20260 citations

Quantum state tomography with time-continuous measurements: reconstruction with resource limitations

Areeya Chantasri, Shengshi Pang, Teerawat Chalermpusitarak +1 more

arXiv:1805.11807Quantum Studies Mathematics and FoundationsMay 27, 201910 citations

Precision-Guaranteed Quantum Tomography

Takanori Sugiyama, Peter S. Turner, Mio Murao

Quantum state tomography is currently the standard tool for verifying that a state prepared in the lab is close to an ideal target state, but up to now there have been no rigorous methods for evaluating the precision of the state preparation in tomographic experiments. We propose a new estimator for quantum state tomography, and prove that the (always physical) estimates will be close to the true prepared state with a high probability. We derive an explicit formula for evaluating how high the probability is for an arbitrary finite-dimensional system and explicitly give the one- and two-qubit cases as examples. This formula applies for any informationally complete sets of measurements, arbitrary finite number of data sets, and general loss functions including the infidelity, the Hilbert-Schmidt, and the trace distances. Using the formula, we can evaluate not only the difference between the estimated and prepared states, but also the difference between the prepared and target states. This is the first result directly applicable to the problem of evaluating the precision of estimation and preparation in quantum tomographic experiments.Quantum state tomography is currently the standard tool for verifying that a state prepared in the lab is close to an ideal target state, but up to now there have been no rigorous methods for evaluating the precision of the state preparation in tomographic experiments. We propose a new estimator for quantum state tomography, and prove that the (always physical) estimates will be close to the true prepared state with a high probability. We derive an explicit formula for evaluating how high the probability is for an arbitrary finite-dimensional system and explicitly give the one- and two-qubit cases as examples. This formula applies for any informationally complete sets of measurements, arbitrary finite number of data sets, and general loss functions including the infidelity, the Hilbert-Schmidt, and the trace distances. Using the formula, we can evaluate not only the difference between the estimated and prepared states, but also the difference between the prepared and target states. This is the first result directly applicable to the problem of evaluating the precision of estimation and preparation in quantum tomographic experiments.

arXiv:1306.4191Physical Review LettersOct 17, 201353 citations

Learning quantum states of continuous-variable systems

Francesco Anna Mele, Antonio Anna Mele, Lennart Bittel +5 more

Quantum measurements are probabilistic and, in general, provide only partial information about the underlying quantum state. Obtaining a full classical description of an unknown quantum state requires the analysis of several different measurements, a task known as quantum-state tomography. Here we analyse the ultimate achievable performance in the tomography of continuous-variable systems, such as bosonic and quantum optical systems. We prove that tomography of these systems is extremely inefficient in terms of time resources, much more so than tomography of finite-dimensional systems such as qubits. Not only does the minimum number of state copies needed for tomography scale exponentially with the number of modes, but, even for low-energy states, it also scales unfavourably with the trace-distance error between the original state and its estimated classical description. On a more positive note, we prove that the tomography of Gaussian states is efficient by establishing a bound on the trace-distance error made when approximating a Gaussian state from knowledge of the first and second moments within a specified error bound. Last, we demonstrate that the tomography of non-Gaussian states prepared through Gaussian unitaries and a few local non-Gaussian evolutions is efficient and experimentally feasible.Quantum measurements are probabilistic and, in general, provide only partial information about the underlying quantum state. Obtaining a full classical description of an unknown quantum state requires the analysis of several different measurements, a task known as quantum-state tomography. Here we analyse the ultimate achievable performance in the tomography of continuous-variable systems, such as bosonic and quantum optical systems. We prove that tomography of these systems is extremely inefficient in terms of time resources, much more so than tomography of finite-dimensional systems such as qubits. Not only does the minimum number of state copies needed for tomography scale exponentially with the number of modes, but, even for low-energy states, it also scales unfavourably with the trace-distance error between the original state and its estimated classical description. On a more positive note, we prove that the tomography of Gaussian states is efficient by establishing a bound on the trace-distance error made when approximating a Gaussian state from knowledge of the first and second moments within a specified error bound. Last, we demonstrate that the tomography of non-Gaussian states prepared through Gaussian unitaries and a few local non-Gaussian evolutions is efficient and experimentally feasible.

Nature PhysicsNov 26, 202513 citations

Quantum-State Tomography and Quantum Communication

Ulf Leonhardt, Igor Jex

Springer eBooksJan 1, 19960 citations

s

-bit program within a given time bound. The learner is required to produce a state that is

ε

-close in trace distance to the target with probability at least

1-δ

. We establish an information-theoretic inference algorithm with round complexity

O(sε^{-2}δ^{-1})

. We further prove a matching lower bound in the relevant parameter regime, even for classical sources. As a second result, we construct a non-i.i.d. state tomography algorithm that outputs a classical description of the conditional state of the next system given past measurement outcomes. Our information-theoretic tomography algorithm achieves round complexity

O(s\min\{2^s,2^n\}ε^{-2}δ^{-1})

, where

n

is the number of qubits received in each round. Finally, we investigate the computational hardness of universal quantum inductive inference under quantum cryptographic assumptions.