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Register a new userSample Efficient Algorithms for Learning Quantum Channels in PAC Model and the Approximate State Discrimination Problem
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We generalize the PAC (probably approximately correct) learning model to the quantum world by generalizing the concepts from classical functions to quantum processes, defining the problem of emph{PAC learning quantum process}, and study its sample complexity. In the problem of PAC learning quantum process, we want to learn an $epsilon$-approximate of an unknown quantum process $c^*$ from a known finite concept class $C$ with probability $1-delta$ using samples ${(x_1,c^*(x_1)),(x_2,c^*(x_2)),dots}$, where ${x_1,x_2, dots}$ are computational basis states sampled from an unknown distribution $D$ and ${c^*(x_1),c^*(x_2),dots}$ are the (possibly mixed) quantum states outputted by $c^*$. The special case of PAC-learning quantum process under constant input reduces to a natural problem which we named as approximate state discrimination, where we are given copies of an unknown quantum state $c^*$ from an known finite set $C$, and we want to learn with probability $1-delta$ an $epsilon$-approximate of $c^*$ with as few copies of $c^*$ as possible. We show that the problem of PAC learning quantum process can be solved with $$Oleft(frac{log|C| + log(1/ delta)} { epsilon^2}right)$$ samples when the outputs are pure states and $$Oleft(frac{log^3 |C|(log |C|+log(1/ delta))} { epsilon^2}right)$$ samples if the outputs can be mixed. Some implications of our results are that we can PAC-learn a polynomial sized quantum circuit in polynomial samples and approximate state discrimination can be solved in polynomial samples even when concept class size $|C|$ is exponential in the number of qubits, an exponentially improvement over a full state tomography.
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$ ewcommand{eps}{varepsilon} $In learning theory, the VC dimension of a concept class $C$ is the most common way to measure its richness. In the PAC model $$ ThetaBig(frac{d}{eps} + frac{log(1/delta)}{eps}Big) $$ examples are necessary and sufficient for a learner to output, with probability $1-delta$, a hypothesis $h$ that is $eps$-close to the target concept $c$. In the related agnostic model, where the samples need not come from a $cin C$, we know that $$ ThetaBig(frac{d}{eps^2} + frac{log(1/delta)}{eps^2}Big) $$ examples are necessary and sufficient to output an hypothesis $hin C$ whose error is at most $eps$ worse than the best concept in $C$. Here we analyze quantum sample complexity, where each example is a coherent quantum state. This model was introduced by Bshouty and Jackson, who showed that quantum examples are more powerful than classical examples in some fixed-distribution settings. However, Atici and Servedio, improved by Zhang, showed that in the PAC setting, quantum examples cannot be much more powerful: the required number of quantum examples is $$ OmegaBig(frac{d^{1-eta}}{eps} + d + frac{log(1/delta)}{eps}Big)mbox{ for all }eta> 0. $$ Our main result is that quantum and classical sample complexity are in fact equal up to constant factors in both the PAC and agnostic models. We give two approaches. The first is a fairly simple information-theoretic argument that yields the above two classical bounds and yields the same bounds for quantum sample complexity up to a $log(d/eps)$ factor. We then give a second approach that avoids the log-factor loss, based on analyzing the behavior of the Pretty Good Measurement on the quantum state identification problems that correspond to learning. This shows classical and quantum sample complexity are equal up to constant factors.
We demonstrate that the ability to estimate the relative sign of an arbitrary $n$-qubit quantum state (with real amplitudes), given only $k$ copies of that state, would yield a $kn$-query algorithm for unstructured search. Thus the quantum sample complexity of sign estimation must be exponential: $Omega(2^{n/2}/n)$. In particular, we show that an efficient procedure for solving the sign estimation problem would allow for a polynomial time solution to the NP-complete problem 3-SAT.
Many quantum machine learning (QML) algorithms that claim speed-up over their classical counterparts only generate quantum states as solutions instead of their final classical description. The additional step to decode quantum states into classical vectors normally will destroy the quantum advantage in most scenarios because all existing tomographic methods require runtime that is polynomial with respect to the state dimension. In this Letter, we present an efficient readout protocol that yields the classical vector form of the generated state, so it will achieve the end-to-end advantage for those quantum algorithms. Our protocol suits the case that the output state lies in the row space of the input matrix, of rank $r$, that is stored in the quantum random access memory. The quantum resources for decoding the state in $ell_2$-norm with $epsilon$ error require $text{poly}(r,1/epsilon)$ copies of the output state and $text{poly}(r, kappa^r,1/epsilon)$ queries to the input oracles, where $kappa$ is the condition number of the input matrix. With our read-out protocol, we completely characterise the end-to-end resources for quantum linear equation solvers and quantum singular value decomposition. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure, which we believe, will be of independent interest.
In this work, we consider optimal state discrimination for a quantum system that interacts with an environment, i.e., states evolve under a quantum channel. We show the conditions on a quantum channel and an ensemble of states such that a measurement for optimal state discrimination is preserved. In particular, we show that when an ensemble of states with equal {it a priori} probabilities is given, an optimal measurement can be preserved over any quantum channel by applying local operations and classical communication, that is, by manipulating the quantum states before and after the channel application. Examples are provided for illustration. Our results can be readily applied to quantum communication protocols over various types of noise.
In this paper, we propose efficient probabilistic algorithms for several problems regarding the autocorrelation spectrum. First, we present a quantum algorithm that samples from the Walsh spectrum of any derivative of $f()$. Informally, the autocorrelation coefficient of a Boolean function $f()$ at some point $a$ measures the average correlation among the values $f(x)$ and $f(x oplus a)$. The derivative of a Boolean function is an extension of autocorrelation to correlation among multiple values of $f()$. The Walsh spectrum is well-studied primarily due to its connection to the quantum circuit for the Deutsch-Jozsa problem. We extend the idea to Higher-order Deutsch-Jozsa quantum algorithm to obtain points corresponding to large absolute values in the Walsh spectrum of a certain derivative of $f()$. Further, we design an algorithm to sample the input points according to squares of the autocorrelation coefficients. Finally we provide a different set of algorithms for estimating the square of a particular coefficient or cumulative sum of their squares.
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