Demonstrating Quantum Advantage with the Simplest Quantum System -- Qubit
Authors: Xiao Yuan, Ke Liu, Luyan Sun, Xiongfeng Ma
Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China.
Along the search for quantum algorithms, the sophisticated quantum speed-up generally arises with delicately designed quantum circuits by manipulating quantum states that contain intricate multipartite correlations. While the essence of quantum correlation originates from coherent superposition of different states, it is natural to expect the essence of quantum advantage to also originate from coherence. This raises a fundamental question: Can quantum advantage be obtained even with the simplest quantum state system, qubit, i.e., superposition of a two level system. The question was answered affirmatively in our recent work published in Physical Review Letters on June 29th .
In our everyday life, a classical coin is called p-coin if it outputs a head and a tail with probability p and 1-p respectively. Given an unknown p-coin, a simple yet interesting problem is to construct an f (o)-coin, where f (p) is a given function of p and f(p)∈[0,1]. For example, when f (p)=1/2, a rather simple but heuristic strategy is given by Von Neumann . Flip the p-coin (p ≠ 0) twice. If the outcomes are the same, start over; otherwise, output the second coin value as the 1/2-coin output. In general, such construction processing is called a Bernoulli factory. Solved by Keane and O’Brien , it says that not all functions can be constructed classically. Generally speaking, a necessary condition for f(p) being constructible is that f(p) ≠ 0 or 1 when p ∈ (0,1). A simple example is f (p) = 4p (1-p), where we have f (1/2)=1.
Figure 1: Classical and quantum coin.
As shown in Fig.1, a p-coin corresponds to a machine that outputs identically mixed qubit states, ρc = p|0⟩⟨0| + (1-p)|1⟩⟨1|, where p∈[0,1]. In general, such unknown p can also be encoded in a quantum way, |p⟩ = √p |0⟩ + √(1-p) |1⟩, which is called by a quoin. As classical coins can always be constructed via a quoin, a natural question is whether the set of quantum constructible functions (via a quantum Bernoulli factory) is strictly larger than the classical set.
Remarkably, Dale et al.  have theoretically proved the necessary and sufficient conditions for quantum Bernoulli factory. Especially for the function f(p) = 4 p (1-p), they proposed a method to construct it by simultaneously measuring two p-quoins. Essentially, entanglement is not necessary for constructing quantum Bernoulli factory. Therefore, we focus on the function f(p) = 4p (1-p) and show the quantum advantage in both theory and experiment with the simplest quantum system.
In practice, we cannot realize exact f(p)-coin due to imperfections, which may cause the realized function classically constructible. However, the number of classical coins N required to construct f(p) generally scales poorly to the inverse of the deviation. Thus, we need to implement high-fidelity state preparation and measurement to reduce the deviation as small as possible in order to faithfully demonstrate the quantum advantage. Superconducting quantum systems have made tremendous progress in the last decade, including a realization of long coherence times, showing great stability with fast and precise qubit manipulations, and demonstrating high-fidelity quantum non-demolition (QND) qubit measurement. Thus, it serves as a perfect candidate for our test.
Figure 2: Experimental setup. (a) Optical image of a transmon qubit located in a trench, which dispersively couples to two 3D Al cavities. (b) Optical image of the single-junction transmon qubit. (c) Scanning electron microscope image of the Josephson junction. (d) Schematic of the device with the main parameters.
The experiment setup is shown in Fig. 2. The necessary high fidelity (~99.6%) and QND qubit detection can be realized with the help of a near-quantum-limited Josephson parametric amplifier [4,5]. A randomized benchmark calibration shows that the single-qubit gate fidelity is about 99.8%, allowing for a highly precise qubit manipulation. Therefore, with the high fidelity state preparation, manipulation and measurement, we are able to achieve fexp (1/2)=0.965. For a special model of the experiment data, we show that more than 105 classical coins are needed for simulating this model, while the average number of quoins for our protocol is about 20.
Our experimental verification sheds light on a fundamental question about what the essential resource for quantum information processing is, which may stimulate the search for more protocols that show quantum advantages without multipartite correlations. Considering the conversion from coherence to multipartite correlation, investigating the power of coherence may also be helpful in understanding the power of multipartite correlation and universal quantum computation.
 Xiao Yuan, Ke Liu, Yuan Xu, Weiting Wang, Yuwei Ma, Fang Zhang, Zhaopeng Yan, R. Vijay, Luyan Sun, Xiongfeng Ma, "Experimental Quantum Randomness Processing Using Superconducting Qubits", Physical Review Letters, 117, 010502 (2016). Abstract.
 J. Von Neumann, "Various Techniques used in connection with random digits", Journal of Research of the National Bureau of Standards -- Applied Mathematics Series, 12, 36 (1951). PDF File.
 M. S. Keane, George L. O’Brien, "A Bernoulli factory", ACM Transactions on Modeling and Computer Simulation, 4, 213 (1994). Abstract.
 Howard Dale, David Jennings, Terry Rudolph, Nature Communications, 6, 8203 (2015). Abstract.
 M. Hatridge, R. Vijay, D.H. Slichter, John Clarke, I. Siddiqi, "Dispersive magnetometry with a quantum limited SQUID parametric amplifier", Physical Review B, 83, 134501 (2011). Abstract.