A key step toward demonstrating a quantum system that can address difficult problems in physics and chemistry will be performing a computation beyond the capabilities of any classical computer, thus achieving so-called quantum supremacy. In this study, we used nine superconducting qubits to demonstrate a promising path toward quantum supremacy. By individually tuning the qubit parameters, we were able to generate thousands of distinct Hamiltonian evolutions and probe the output probabilities. The measured probabilities obey a universal distribution, consistent with uniformly sampling the full Hilbert space. As the number of qubits increases, the system continues to explore the exponentially growing number of states. Extending these results to a system of 50 qubits has the potential to address scientific questions that are beyond the capabilities of any classical computer. A programmable quantum system consisting of merely 50 to 100 qubits could have a marked impact on scientific research. Although such a platform is naturally suited to address problems in quantum chemistry and materials science (1-4), applications extend to fields as diverse as classical dynamics (5) and computer science (6-9). An important milestone on the path toward realizing these applications will be the demonstration of an algorithm that exceeds the capabilities of any classical computer, thus achieving quantum supremacy (10). Sampling problems are an iconic example of algorithms designed specifically for this purpose (11) (12) (13) (14) . A successful demonstration of quantum supremacy would prove that engineered quantum systems, although still in their infancy, can outperform the most advanced classical computers. Consider a system of coupled qubits whose dynamics uniformly explore all accessible states over time. The complexity of simulating this evolution on a classical computer is easy to understand and quantify. Because every state is equally important, it is not possible to simplify the problem by using a smaller truncated state-space. The complexity is then simply given by how much classical memory it takes to store the state vector. Storing the state of a 46-qubit system requires nearly a petabyte of memory and is at the limit of the most powerful computers (14, 15) . Sampling from the output probabilities of such a system would therefore constitute a clear demonstration of quantum supremacy. Note that this is an upper bound on only the number of qubits required-other constraints, such as computation time, may place practical limitations on even smaller system sizes. In this study, we experimentally illustrate a blueprint for demonstrating quantum supremacy. We present data characterizing two basic ingredients required for any supremacy experiment: complexity and fidelity. First, we show that the qubits can quasi-uniformly explore the Hilbert space, providing an experimental indication of algorithm complexity [see (16) for a formal discussion of computational complexity]. Next, we compare the measurement results with the expected behavior and show that the algorithm can be implemented with high fidelity. Experiments probing complexity and fidelity provide a foundation for demonstrating quantum supremacy. The more control a quantum platform offers, the easier it is to embed diverse applications. For this reason, we have developed superconducting gmon qubits, which are based on transmon qubits but have tunable frequencies and tunable interactions (Fig. 1A) . The nine-qubit device consists of three distinct sections: control (bottom), qubits (center) and readout (top). A detailed circuit diagram is provided in (16). Each of our gmon qubits can be thought of as a nonlinear oscillator. The Hamiltonian for the device is given by wheren is the number operator andâ † (â) is the raising (lowering) operator. The qubit frequency sets the coefficient d i , the nonlinearity sets h i , and the nearest-neighbor coupling sets g i . The two lowest energy levels (j0i and j1i) form the qubit subspace. The higher energy levels of the qubits, although only virtually occupied, substantially modify the dynamics. In the absence of higher levels, this model maps to free particles and can be simulated efficiently (16). The inclusion of higher levels effectively introduces an interaction and allows for the occurrence of complex dynamics. In Fig. 1, B and C, we outline the experimental procedure and provide two instances of the raw output data. Figure 1B shows a five-qubit example of the pulses used to control the qubits. First, the system is initialized (red) by placing two of the qubits in the excited state; e.g., j00101i. The dynamics result from fixing the qubit frequencies (orange) and simultaneously ramping all of the nearest-neighbor interactions on and then off (green). The shape of the coupling pulse is chosen to minimize leakage out of the qubit subspace (17). After the evolution, we simultaneously measure the state of every qubit. Each measurement results in a single output state, such as j10010i; the experiment is repeated many times to estimate the probability of every possible output state. We then carry out this procedure for randomly chosen values of the qubit frequencies, the coupler pulse lengths, and the coupler pulse heights. The probabilities of the various output states are shown in Fig. 1C for two instances of the evolution after 10 coupler pulses (cycles). The height of each bar represent the probability with which that output state appeared in the experiments. The Hamiltonian in Eq. 1 conserves the total number of excitations. This means that if we start in a state with half of the qubits excited, we should also end in a state with half of the qubits excited. However, most experimental errors do not obey this symmetry, allowing us to identify and remove erroneous outcomes. Although this symmetry helps to reduce the impact of errors, it slightly reduces the size of the Hilbert space. For N qubits, the number of states is given by the permutations of N/2 excitations in N qubits and is approximately 2 N = ffiffiffiffi N p . As an example, a 64-qubit system would access~2 61 states under our protocol. Although the measured probabilities appear largely random, they provide important insight into the quantum dynamics of the system. A key feature of these data sets are the rare, taller-thanaverage peaks, which are analogous to the highintensity regions of a laser's speckle pattern. These highly likely states serve as a fingerprint of the underlying evolution and provide a means for verifying that the desired evolution was properly generated. The distribution of these probabilities provides evidence that the dynamics coherently and uniformly explore the Hilbert space. In Fig. 2 , we use the measured probabilities to show that the dynamics uniformly explore the Hilbert space for experiments carried out with
Google proposes a plan to construct random circuits to demonstrate quantum supremacy at the pulse level REF .
A blueprint for demonstrating quantum supremacy with superconducting qubits
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