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

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 106 ◽  
Author(s):  
Tomoyuki Morimae ◽  
Yuki Takeuchi ◽  
Harumichi Nishimura
Keyword(s):  
Quantum Computing ◽  
Polynomial Time ◽  
Probability Distributions ◽  
Quantum Circuits ◽  
Universal Models ◽  
Reversible Circuit ◽  
Universal Quantum ◽  
Quantum Polynomial ◽  
Output Probability ◽  
Probabilistic Polynomial Time

We introduce a simple sub-universal quantum computing model, which we call the Hadamard-classical 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. 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.

scholarly journals Fine-grained quantum computational supremacy

2019 ◽  
Vol 19 (13&14) ◽  
pp. 1089-1115
Author(s):  
Tomoyuki Morimae ◽  
Suguru Tamaki
Keyword(s):  
Quantum Computing ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Probability Distributions ◽  
Fine Grained ◽  
Multiplicative Error ◽  
Universal Quantum ◽  
The One ◽  
Output Probability ◽  
Computing Models

(pp1089-1115) Tomoyuki Morimae and Suguru Tamaki doi: https://doi.org/10.26421/QIC19.13-14-2 Abstracts: Output probability distributions of several sub-universal quantum computing models cannot be classically efficiently sampled unless some unlikely consequences occur in classical complexity theory, such as the collapse of the polynomial-time hierarchy. These results, so called quantum supremacy, however, do not rule out possibilities of super-polynomial-time classical simulations. In this paper, we study ``fine-grained" version of quantum supremacy that excludes some exponential-time classical simulations. First, we focus on two sub-universal models, namely, the one-clean-qubit model (or the DQC1 model) and the HC1Q model. Assuming certain conjectures in fine-grained complexity theory, we show that for any a>0 output probability distributions of these models cannot be classically sampled within a constant multiplicative error and in 2^{(1-a)N+o(N)} time, where N is the number of qubits. Next, we consider universal quantum computing. For example, we consider quantum computing over Clifford and T gates, and show that under another fine-grained complexity conjecture, output probability distributions of Clifford-T quantum computing cannot be classically sampled in 2^{o(t)} time within a constant multiplicative error, where t is the number of T gates.

scholarly journals Achieving quantum supremacy with sparse and noisy commuting quantum computations

Quantum ◽  
2017 ◽  
Vol 1 ◽  
pp. 8 ◽  
Author(s):  
Michael J. Bremner ◽  
Ashley Montanaro ◽  
Dan J. Shepherd
Keyword(s):  
Probability Distribution ◽  
Error Correction ◽  
Polynomial Time ◽  
Quantum Circuits ◽  
Square Lattice ◽  
Constant Amount ◽  
Small Constant ◽  
Variation Distance ◽  
Quantum Polynomial ◽  
Output Probability

The class of commuting quantum circuits known as IQP (instantaneous quantum polynomial-time) has been shown to be hard to simulate classically, assuming certain complexity-theoretic conjectures. Here we study the power of IQP circuits in the presence of physically motivated constraints. First, we show that there is a family of sparse IQP circuits that can be implemented on a square lattice of n qubits in depth O(sqrt(n) log n), and which is likely hard to simulate classically. Next, we show that, if an arbitrarily small constant amount of noise is applied to each qubit at the end of any IQP circuit whose output probability distribution is sufficiently anticoncentrated, there is a polynomial-time classical algorithm that simulates sampling from the resulting distribution, up to constant accuracy in total variation distance. However, we show that purely classical error-correction techniques can be used to design IQP circuits which remain hard to simulate classically, even in the presence of arbitrary amounts of noise of this form. These results demonstrate the challenges faced by experiments designed to demonstrate quantum supremacy over classical computation, and how these challenges can be overcome.

scholarly journals Additive-error fine-grained quantum supremacy

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 329
Author(s):  
Tomoyuki Morimae ◽  
Suguru Tamaki
Keyword(s):  
Quantum Computing ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Boolean Circuits ◽  
Exponential Time ◽  
Fine Grained ◽  
Additive Error ◽  
Universal Quantum ◽  
The One ◽  
Computing Models

It is known that several sub-universal quantum computing models, such as the IQP model, the Boson sampling model, the one-clean qubit model, and the random circuit model, cannot be classically simulated in polynomial time under certain conjectures in classical complexity theory. Recently, these results have been improved to ``fine-grained" versions where even exponential-time classical simulations are excluded assuming certain classical fine-grained complexity conjectures. All these fine-grained results are, however, about the hardness of strong simulations or multiplicative-error sampling. It was open whether any fine-grained quantum supremacy result can be shown for a more realistic setup, namely, additive-error sampling. In this paper, we show the additive-error fine-grained quantum supremacy (under certain complexity assumptions). As examples, we consider the IQP model, a mixture of the IQP model and log-depth Boolean circuits, and Clifford+T circuits. Similar results should hold for other sub-universal models.

scholarly journals Quantum computing, postselection, and probabilistic polynomial-time

2005 ◽  
Vol 461 (2063) ◽  
pp. 3473-3482 ◽  
Author(s):  
Scott Aaronson
Keyword(s):  
Quantum Mechanics ◽  
Quantum Computing ◽  
Polynomial Time ◽  
Quantum Computer ◽  
Complexity Class ◽  
Complete Problems ◽  
Probabilistic Polynomial Time

I study the class of problems efficiently solvable by a quantum computer, given the ability to ‘postselect’ on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or probabilistic polynomial-time. Using this result, I show that several simple changes to the axioms of quantum mechanics would let us solve PP-complete problems efficiently. The result also implies, as an easy corollary, a celebrated theorem of Beigel, Reingold and Spielman that PP is closed under intersection, as well as a generalization of that theorem due to Fortnow and Reingold. This illustrates that quantum computing can yield new and simpler proofs of major results about classical computation.

scholarly journals Pseudo-dimension of quantum circuits

2020 ◽  
Vol 2 (2) ◽  
Author(s):  
Matthias C. Caro ◽  
Ishaun Datta
Keyword(s):  
Probability Distributions ◽  
Expressive Power ◽  
Upper Bounds ◽  
Quantum Circuits ◽  
Polynomial Size ◽  
Probabilistic Concept ◽  
Circuit Depth ◽  
Concept Classes ◽  
Circuit Output ◽  
Output Probability

AbstractWe characterize the expressive power of quantum circuits with the pseudo-dimension, a measure of complexity for probabilistic concept classes. We prove pseudo-dimension bounds on the output probability distributions of quantum circuits; the upper bounds are polynomial in circuit depth and number of gates. Using these bounds, we exhibit a class of circuit output states out of which at least one has exponential gate complexity of state preparation, and moreover demonstrate that quantum circuits of known polynomial size and depth are PAC-learnable.

scholarly journals EXACT NON-IDENTITY CHECK IS NQP-COMPLETE

2010 ◽  
Vol 08 (05) ◽  
pp. 807-819
Author(s):  
YU TANAKA
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Decision Problem ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Quantum Gates ◽  
Unitary Operation ◽  
Classical Description ◽  
Quantum Polynomial ◽  
Equivalence Check

To understand quantum gate array complexity, we define a problem named exact non-identity check, which is a decision problem to determine whether a given classical description of a quantum circuit is strictly equivalent to the identity or not. We show that the computational complexity of this problem is non-deterministic quantum polynomial-time (NQP)-complete. As corollaries, it is derived that exact non-equivalence check of two given classical descriptions of quantum circuits is also NQP-complete and that minimizing the number of quantum gates for a given quantum circuit without changing the implemented unitary operation is NQP-hard.

scholarly journals Rational proofs for quantum computing

2020 ◽  
Vol 20 (3&4) ◽  
pp. 181-193
Author(s):  
Tomoyuki Morimae ◽  
Harumichi Harumichi Nishimura
Keyword(s):  
Quantum Computing ◽  
Polynomial Time ◽  
Open Problem ◽  
Correct Solution ◽  
Expectation Value ◽  
Reward Function ◽  
Single Qubit ◽  
Quantum Technology ◽  
Qubit States ◽  
Probabilistic Polynomial Time

It is an open problem whether a classical client can delegate quantum computing to an efficient remote quantum server in such a way that the correctness of quantum computing is somehow guaranteed. Several protocols for verifiable delegated quantum computing have been proposed, but the client is not completely free from any quantum technology: the client has to generate or measure single-qubit states. In this paper, we show that the client can be completely classical if the server is rational (i.e., economically motivated), following the ``rational proofs" framework of Azar and Micali. More precisely, we consider the following protocol. The server first sends the client a message allegedly equal to the solution of the problem that the client wants to solve. The client then gives the server a monetary reward whose amount is calculated in classical probabilistic polynomial-time by using the server's message as an input. The reward function is constructed in such a way that the expectation value of the reward (the expectation over the client's probabilistic computing) is maximum when the server's message is the correct solution to the problem. The rational server who wants to maximize his/her profit therefore has to send the correct solution to the client.

scholarly journals Commuting quantum circuits with few outputs are unlikely to be classically simulatable

2016 ◽  
Vol 16 (3&4) ◽  
pp. 251-270 ◽  
Author(s):  
Yasuhiro Takahashi ◽  
Seiichiro Tani ◽  
Takeshi Yamazaki ◽  
Kazuyuki Tanaka
Keyword(s):  
Probability Distribution ◽  
Polynomial Time ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Polynomial Hierarchy ◽  
The Third ◽  
Additive Error ◽  
N Input ◽  
Output Probability ◽  
Small Additive

We study the classical simulatability of commuting quantum circuits with n input qubits and O(log n) output qubits, where a quantum circuit is classically simulatable if its output probability distribution can be sampled up to an exponentially small additive error in classical polynomial time. Our main result is that there exists a commuting quantum circuit that is not classically simulatable unless the polynomial hierarchy collapses to the third level. This is the first formal evidence that a commuting quantum circuit is not classically simulatable even when the number of output qubits is O(log n). Then, we consider a generalized version of the circuit and clarify the condition under which it is classically simulatable. Lastly, using a proof similar to that of the main result, we provide an evidence that a slightly extended Clifford circuit is not classically simulatable.

scholarly journals On statistically-secure quantum homomorphic encryption

2018 ◽  
Vol 18 (9&10) ◽  
pp. 785-794
Author(s):  
Ching-Yi Lai ◽  
Kai-Min Chung
Keyword(s):  
Polynomial Time ◽  
Homomorphic Encryption ◽  
Quantum Circuits ◽  
It Security ◽  
Encryption Scheme ◽  
Information Theoretic ◽  
Information Theoretic Security ◽  
Quantum Polynomial ◽  
The One

Homomorphic encryption is an encryption scheme that allows computations to be evaluated on encrypted inputs without knowledge of their raw messages. Recently Ouyang et al. constructed a quantum homomorphic encryption (QHE) scheme for Clifford circuits with statistical security (or information-theoretic security (IT-security)). It is desired to see whether an information-theoretically-secure (ITS) quantum FHE exists. If not, what other nontrivial class of quantum circuits can be homomorphically evaluated with IT-security? We provide a limitation for the first question that an ITS quantum FHE necessarily incurs exponential overhead. As for the second one, we propose a QHE scheme for the instantaneous quantum polynomial-time (IQP) circuits. Our QHE scheme for IQP circuits follows from the one-time pad.

scholarly journals Classical simulation complexity of extended Clifford circuits

2014 ◽  
Vol 14 (7&8) ◽  
pp. 633-648
Author(s):  
Richard Jozsa ◽  
Marrten Van den Nest
Keyword(s):  
Quantum Computing ◽  
Physical Significance ◽  
Quantum Circuits ◽  
Quantum Processes ◽  
Computing Power ◽  
Quantum Operations ◽  
Classical Simulation ◽  
Universal Quantum ◽  
General Product ◽  
Computational Basis

Clifford gates are a winsome class of quantum operations combining mathematical elegance with physical significance. The Gottesman-Knill theorem asserts that Clifford computations can be classically efficiently simulated but this is true only in a suitably restricted setting. Here we consider Clifford computations with a variety of additional ingredients: (a) strong vs. weak simulation, (b) inputs being computational basis states vs. general product states, (c) adaptive vs. non-adaptive choices of gates for circuits involving intermediate measurements, (d) single line outputs vs. multi-line outputs. We consider the classical simulation complexity of all combinations of these ingredients and show that many are not classically efficiently simulatable (subject to common complexity assumptions such as P not equal to NP). Our results reveal a surprising proximity of classical to quantum computing power viz. a class of classically simulatable quantum circuits which yields universal quantum computation if extended by a purely classical additional ingredient that does not extend the class of quantum processes occurring.

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