Clifford Circuits can be Properly PAC Learned if and only if $\textsf{RP}=\textsf{NP}$

• URL: http://arxiv.org/abs/2204.06638v4
• Date: Tue, 6 Jun 2023 00:15:47 GMT
• Title: Clifford Circuits can be Properly PAC Learned if and only if $\textsf{RP}=\textsf{NP}$
• Authors: Daniel Liang
• Abstract summary: We show that proper learning of Clifford circuits is hard for classical learners unless $textsfRP = textsfNP$. We also show that an efficient proper quantum learner exists if and only if $textsfNP subseteq textsfRQP$.
• Score: 1.14219428942199
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Given a dataset of input states, measurements, and probabilities, is it possible to efficiently predict the measurement probabilities associated with a quantum circuit? Recent work of Caro and Datta (2020) studied the problem of PAC learning quantum circuits in an information theoretic sense, leaving open questions of computational efficiency. In particular, one candidate class of circuits for which an efficient learner might have been possible was that of Clifford circuits, since the corresponding set of states generated by such circuits, called stabilizer states, are known to be efficiently PAC learnable (Rocchetto 2018). Here we provide a negative result, showing that proper learning of CNOT circuits is hard for classical learners unless $\textsf{RP} = \textsf{NP}$. As the classical analogue and subset of Clifford circuits, this naturally leads to a hardness result for Clifford circuits as well. Additionally, we show that if $\textsf{RP} = \textsf{NP}$ then there would exist efficient proper learning algorithms for CNOT and Clifford circuits. By similar arguments, we also find that an efficient proper quantum learner for such circuits exists if and only if $\textsf{NP} \subseteq \textsf{RQP}$.

Related papers

• Learning shallow quantum circuits [7.411898489476803]
  We present an algorithm for learning the description of any unknown $n$-qubit shallow quantum circuit $U$. We also provide a classical algorithm for learning the description of any unknown $n$-qubit state $lvert psi rangle$. Our approach uses a quantum circuit representation based on local inversions and a technique to combine these inversions.
  arXiv  Detail & Related papers  (2024-01-18T16:05:00Z)
• On the Pauli Spectrum of QAC0 [2.3436632098950456]
  We conjecture that the Pauli spectrum of $mathsfQAC0$ satisfies low-degree concentration. We obtain new circuit lower bounds and learning results as applications.
  arXiv  Detail & Related papers  (2023-11-16T07:25:06Z)
• Simulation of IBM's kicked Ising experiment with Projected Entangled Pair Operator [71.10376783074766]
  We perform classical simulations of the 127-qubit kicked Ising model, which was recently emulated using a quantum circuit with error mitigation. Our approach is based on the projected entangled pair operator (PEPO) in the Heisenberg picture. We develop a Clifford expansion theory to compute exact expectation values and use them to evaluate algorithms.
  arXiv  Detail & Related papers  (2023-08-06T10:24:23Z)
• Learning efficient decoders for quasi-chaotic quantum scramblers [3.823356975862005]
  We show that one can retrieve the scrambled information even without any previous knowledge of the scrambler. A classical decoder can retrieve with fidelity one all the information scrambled by a random unitary. Results show that one can learn the salient properties of quantum unitaries in a classical form.
  arXiv  Detail & Related papers  (2022-12-21T20:19:53Z)
• Classically Simulating Quantum Supremacy IQP Circuits trough a Random Graph Approach [0.0]
  A good candidate for demonstrating quantum supremacy with random circuit sampling is to use emphIQP circuits. We introduce improved techniques for classically simulating random IQP circuits. We estimate that 70-qubit circuits are within reach for a large computing cluster.
  arXiv  Detail & Related papers  (2022-12-16T17:38:42Z)
• Iterative Qubit Coupled Cluster using only Clifford circuits [52.77024349608834]
  We draw attention to a variant of the iterative qubit coupled cluster (iQCC) method that only uses Clifford circuits. This method is useful for near-term variational quantum algorithm applications as it generates good initial parameters. It may also be useful beyond the NISQ era to create short-depth Clifford pre-optimized circuits.
  arXiv  Detail & Related papers  (2022-11-18T20:31:10Z)
• Quantum Depth in the Random Oracle Model [57.663890114335736]
  We give a comprehensive characterization of the computational power of shallow quantum circuits combined with classical computation. For some problems, the ability to perform adaptive measurements in a single shallow quantum circuit is more useful than the ability to perform many shallow quantum circuits without adaptive measurements.
  arXiv  Detail & Related papers  (2022-10-12T17:54:02Z)
• A single $T$-gate makes distribution learning hard [56.045224655472865]
  This work provides an extensive characterization of the learnability of the output distributions of local quantum circuits. We show that for a wide variety of the most practically relevant learning algorithms -- including hybrid-quantum classical algorithms -- even the generative modelling problem associated with depth $d=omega(log(n))$ Clifford circuits is hard.
  arXiv  Detail & Related papers  (2022-07-07T08:04:15Z)
• Learning quantum circuits of some $T$ gates [10.609715843964263]
  It has been unknown how to handle circuits beyond the Clifford group. We show that the output state of a $T$-depth one circuit textitof full $T$-rank can be represented by a stabilizer pseudomixture.
  arXiv  Detail & Related papers  (2021-06-23T16:43:01Z)
• On the realistic worst case analysis of quantum arithmetic circuits [69.43216268165402]
  We show that commonly held intuitions when designing quantum circuits can be misleading. We show that reducing the T-count can increase the total depth. We illustrate our method on addition and multiplication circuits using ripple-carry.
  arXiv  Detail & Related papers  (2021-01-12T21:36:16Z)
• Machine Learning Optimization of Quantum Circuit Layouts [63.55764634492974]
  We introduce a quantum circuit mapping, QXX, and its machine learning version, QXX-MLP. The latter infers automatically the optimal QXX parameter values such that the layed out circuit has a reduced depth. We present empiric evidence for the feasibility of learning the layout method using approximation.
  arXiv  Detail & Related papers  (2020-07-29T05:26:19Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.