Classical verification of quantum circuits containing few basis changes

Demarie, Tommaso F.; Ouyang, Yingkai; Fitzsimons, Joseph F.

doi:10.1103/physreva.97.042319

Cited by 9 publications

(7 citation statements)

References 42 publications

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“…eqn. (22). Then we apply a variational circuit W ( θ) to the initial state that depends on some variational parameters θ, c.f.…”

Section: Variational Circuit Classifiersmentioning

confidence: 99%

“…The exact evaluation of the inner-product between two states generated from a similar circuit with only a single diagonal layer U Φ( x) is #P -hard [21]. Nonetheless, in the experimentally relevant context of additive error approximation, simulation of a single layer preparation circuit can be achieved efficiently classically by uniform sampling [22]. We conjecture that the evaluation of inner products generated from circuits with two basis changes and diagonal gates up to additive error to be hard, c.f.…”

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confidence: 99%

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Supervised learning with quantum-enhanced feature spaces

Havlíček

Córcoles

Temme

et al. 2019

Nature

1,643

1,546

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Machine learning and quantum computing are two technologies each with the potential for altering how computation is performed to address previously untenable problems. Kernel methods for machine learning are ubiquitous for pattern recognition, with support vector machines (SVMs) being the most well-known method for classification problems. However, there are limitations to the successful solution to such problems when the feature space becomes large, and the kernel functions become computationally expensive to estimate. A core element to computational speed-ups afforded by quantum algorithms is the exploitation of an exponentially large quantum state space through controllable entanglement and interference.Here, we propose and experimentally implement two novel methods on a superconducting processor. Both methods represent the feature space of a classification problem by a quantum state, taking advantage of the large dimensionality of quantum Hilbert space to obtain an enhanced solution. One method, the quantum variational classifier builds on [1, 2] and operates through using a variational quantum circuit to classify a training set in direct analogy to conventional SVMs. In the second, a quantum kernel estimator, we estimate the kernel function and optimize the classifier directly. The two methods present a new class of tools for exploring the applications of noisy intermediate scale quantum computers [3] to machine learning.The intersection between machine learning and quantum computing has been dubbed quantum machine learning, and has attracted considerable attention in recent years [4][5][6]. This has led to a number of recently proposed quantum algorithms [1,2,[7][8][9]. Here, we present a quantum algorithm that has the potential to run on near-term quantum devices. A natural class of algorithms for such noisy devices are short-depth circuits, which are amenable to error-mitigation techniques that reduce the effect of decoherence [10,11]. There are convincing arguments that indicate that even very sim- ple circuits are hard to simulate by classical computational means [12,13]. The algorithm we propose takes on the original problem of supervised learning: the construction of a classifier. For this problem, we are given data from a training set T and a test set S of a subset Ω ⊂ R d . Both are assumed to be labeled by a map m : T ∪ S → {+1, −1} unknown to the algorithm. The training algorithm only receives the labels of the training data T . The goal is to infer an approximate map on the test setm : S → {+1, −1} such that it agrees with high probability with the true map m( s) =m( s) on the test data s ∈ S. For such a learning task to be meaningful it is assumed that there is a correlation between the labels given for training and the true map. A classical approach to constructing an approximate labeling function uses socalled support vector machines (SVMs) [14]. The data gets mapped non-linearly to a high dimensional space, the feature space, where a hyperplane is constructed to separate the labeled samples. ...

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“…eqn. (22). Then we apply a variational circuit W ( θ) to the initial state that depends on some variational parameters θ, c.f.…”

Section: Variational Circuit Classifiersmentioning

confidence: 99%

mentioning

confidence: 99%

Supervised learning with quantum-enhanced feature spaces

Havlíček

Córcoles

Temme

et al. 2019

Nature

1,643

1,546

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show abstract

“…Finally, it was suggested in Ref. [32] that a problem of deciding whether there exist some results that occur with high probability or not for circuits in FH 2 has a Merlin-Arthur system with quantum polynomial-time Merlin.…”

Section: It Has Been Shown Recently That the Classical Verification Omentioning

confidence: 99%

Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy

Morimae

Takeuchi

Nishimura

2018

Quantum

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We introduce a simple sub-universal quantum computing model, which we call the Hadamardclassical circuit with one-qubit (HC1Q) model. It consists of a classical reversible circuit sandwiched by two layers of Hadamard gates, and therefore it is in the second level of the Fourier hierarchy [1].We show that output probability distributions of the HC1Q model cannot be classically efficiently sampled within a multiplicative error unless the polynomial-time hierarchy collapses to the second level. The proof technique is different from those used for previous sub-universal models, such as IQP, Boson Sampling, and DQC1, and therefore the technique itself might be useful for finding other sub-universal models that are hard to classically simulate. We also study the classical verification of quantum computing in the second level of the Fourier hierarchy. To this end, we define a promise problem, which we call the probability distribution distinguishability with maximum norm (PDD-Max). It is a promise problem to decide whether output probability distributions of two quantum circuits are far apart or close. We show that PDD-Max is BQP-complete, but if the two circuits are restricted to some types in the second level of the Fourier hierarchy, such as the HC1Q model or the IQP model, PDD-Max has a Merlin-Arthur system with quantum polynomial-time Merlin and classical probabilistic polynomial-time Arthur. * Electronic address: tomoyuki.morimae@yukawa.kyoto-u.ac.jp † Electronic address: takeuchi.yuki@lab.ntt.co.jp ‡ Electronic address: hnishimura@is.nagoya-u.ac.jp lating the IQP model. Other sub-universal models that exhibit similar quantum supremacy are also known, such as the depth four model [6], the Boson sampling model [7], the DQC1 model [8][9][10][11][12], the Fourier sampling model [13], the conjugated Clifford model [14], and the random circuit model [15].The first main result of the present paper is to add another simple model in FH 2 to the above list of sub-universal models that exhibit quantum supremacy. We define the Hadamard-classical circuit with one-qubit (HC1Q) model as follows:Definition 2 (HC1Q) The Hadamard-classical circuit with one-qubit (HC1Q) model on N qubits is the following quantum computing model (see Fig. 1(a)):

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“…As another example, the completely classical client can efficiently delegate it to multiple quantum servers who share entangled states but cannot communicate with each other [18,[21][22][23][24][25][26][27][28]. It is also known that some problems in BQP can be efficiently verified by interactions between the classical client and the quantum server [29][30][31][32]. Examples of such verifiable problems are integer factorization, the recursive Fourier sampling [29], promise problems related to output probability distributions of quantum circuits in the second level of the Fourier hierarchy [30,31], and the calculation of the order of solvable groups [32].…”

Section: Introduction a Backgroundmentioning

confidence: 99%

“…It is also known that some problems in BQP can be efficiently verified by interactions between the classical client and the quantum server [29][30][31][32]. Examples of such verifiable problems are integer factorization, the recursive Fourier sampling [29], promise problems related to output probability distributions of quantum circuits in the second level of the Fourier hierarchy [30,31], and the calculation of the order of solvable groups [32]. Furthermore, it has recently been shown that if the learning-with-errors (LWE) problem is hard for polynomial-time quantum computing, the classical client can delegate verifiable universal quantum computing to the single quantum server whose computational power is bounded by BQP 1 even in the malicious FIG.…”

Section: Introduction a Backgroundmentioning

confidence: 99%

Sumcheck-based delegation of quantum computing to rational server

Takeuchi,

Morimae,

Tani

2019

Preprint

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Delegated quantum computing enables a client with a weak computational power to delegate quantum computing to a remote quantum server in such a way that the integrity of the server is efficiently verified by the client. Recently, a new model of delegated quantum computing has been proposed, namely, rational delegated quantum computing. In this model, after the client interacts with the server, the client pays a reward to the server depending on the server's messages and the client's random bits. The rational server sends messages that maximize the expected value of the reward. It is known that the classical client can delegate universal quantum computing to the rational quantum server in one round. In this paper, we propose novel one-round rational delegated quantum computing protocols by generalizing the classical rational sumcheck protocol. An advantage of our protocols is that they are gate-set independent: the construction of the previous rational protocols depends on gate sets, while our sumcheck technique can be easily realized with any local gate set (each of whose elementary gates can be specified with a polynomial number of bits). Furthermore, as with the previous protocols, our reward function satisfies natural requirements (non-negative, upper-bounded by a constant, and its maximum expected value is lower-bounded by a constant). We also discuss the reward gap. Simply speaking, the reward gap is a minimum loss on the expected value of the server's reward incurred by the server's behavior that makes the client accept an incorrect answer. The reward gap therefore should be large enough to incentivize the server to behave optimally. Although our sumcheck-based protocols have only exponentially small reward gaps as with the previous protocols, we show that a constant reward gap can be achieved if two non-communicating but entangled rational servers are allowed. We also discuss that a single rational server is sufficient under the (widely-believed) assumption that the learning-with-errors problem is hard for polynomial-time quantum computing. Apart from these results, we show, under a certain condition, the equivalence between rational and ordinary delegated quantum computing protocols. Based on this equivalence, we give a reward-gap amplification method.

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Classical verification of quantum circuits containing few basis changes

Cited by 9 publications

References 42 publications

Supervised learning with quantum-enhanced feature spaces

Supervised learning with quantum-enhanced feature spaces

Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy

Sumcheck-based delegation of quantum computing to rational server

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