Measuring the Rényi entropy of a two-site Fermi-Hubbard model on a trapped ion quantum computer

Abstract

The efficient simulation of correlated quantum systems is the most promising near-term application of quantum computers. Here, we present a measurement of the second Renyi entropy of the ground state of the two-site Fermi-Hubbard model on a 5-qubit programmable quantum computer based on trapped ions. Our work illustrates the extraction of a non-linear characteristic of a quantum state using a controlled-swap gate acting on two copies of the state. This scalable measurement of entanglement on a universal quantum computer will, with more qubits, provide insights into many-body quantum systems that are impossible to simulate on classical computers.One of the striking differences between classical and quantum systems is the phenomenon of entanglement. Analyzing large entangled states is of considerable interest for quantum computing applications. This is particularly relevant to quantum chemistry and materials science simulations involving interacting fermions [1,2], small versions of which have been simulated on few-qubit quantum computers [3][4][5]. Recently, a quantum algorithm was developed to construct the entanglement spectrum of an arbitrary wave function prepared on a quantum computer via measurement of the Renyi entropies [6]. In this Letter we measure the second Renyi entropy in a 5-qubit circuit by implementing a controlled-swap (C-Swap) gate, and mitigate experimental errors by exploiting the symmetry properties of this gate. We note that previous measurements of the Renyi entropy such as [7] were not implemented on universal machines and may not be easily generalizable to arbitrary Hamiltonians or scalable to larger systems.For a many-body quantum system ideally described by the state |Ψ and composed of two subsystems A and B, the nth Renyi entropy is given by S n = 1 1−n log(R n ), whereis the trace of the nth power of the reduced density matrix ρ A = Tr B (|Ψ Ψ|). For non-zero entanglement we have R 2 < 1, which has the same universality properties as the von Neumann entropy S = − Tr(ρ A log(ρ A )).Both are measures of the entanglement between A and B, and provide valuable information about the underlying physics of the system. For example, the Renyi entropy can be used to distinguish many-body localized states from thermalized states [8-12] through their time dependence and dimensional scaling law [13], and to study topological order [14,15] and quantum critical systems [16].The system under investigation for this work is the two-site Fermi-Hubbard model, which describes interacting electrons on a lattice [17,18]. Despite its simplicity, it has been postulated as a model for complex phenomena such as high-temperature superconductivity. Since its behavior in the thermodynamic limit remains inaccessible to classical numerical techniques, it has become a prime candidate for simulation by quantum computers [19,20]. Our work consists of several co-designed theoretical and experimental steps. First, we find an efficient mapping from the electronic problem to the qubit space. Second, we develop a circuit for...

Interference, spectral momentum correlations, entanglement, and Bell inequality for a trapped interacting ultracold atomic dimer: Analogies with biphoton interferometry

“…V, principal interest in entanglement, aims at uncovering fundamentals of the quantum world, principally the non-intuitive notion of non-locality (which contradicts local realism), stating that: no physical object has distinctive individual properties that completely define it as an independent entity, and that the result of measurement on one system is not independent of measurements and/or operations performed on another, spatially separate, systems. Entanglement has been used in many-body theoretical and experimantal investigations of correlations, quantum magnetism, and quantum phase transitions in condensed-matter physics [94][95][96][97][98][99], manyparticle ultracold atomic systems trapped in optical lattices [23,24,100,101], atomic and molecular systems [102], and even in biological structures [103][104][105][106][107][108][109][110]. Moreover, since the discovery of Shor's factoring algorithm [111] which is anchored in entanglement, and the consideration of a quantum-gate mechanism based on electron spins in coupled semiconductor quantum dots, which can be used as a general source of spin entanglement in quantum computers [112], there has been a growing interest in entanglement in the burgeoning areas of quantum information, quantum cryptography, and quantum teleportation.…”

Noise resilience of variational quantum compiling

“…Clearly, error mitigation techniques will be necessary to make use of NISQ devices. Several promising error mitigation strategies have recently emerged, including zero-noise extrapolation [2], quasi-probability decomposition [2], post-selection [3,4], noise-aware compiling [5], and machine learning for circuit-depth compression [6]. Let us consider two other strategies for error mitigation in what follows.…”

Probing Rényi entanglement entropy via randomized measurements

“…Thus, a measurement of the entropy of the whole system, as well as of its subsystems, provides information about the entanglement contained within the system. Additionally, a measurement of the entropy of the total state ρ provides the opportunity to verify the overall coherence of the system, as for pure quantum states S (2) (ρ) = 0.Recently, a protocol to directly measure the second-order Rényi entropy, S (2) , has been demonstrated, requiring collective measurements to be made on two identical copies ρ of a quantum system (15)(16)(17)(18). In (17), that protocol was used to study entanglement growth and thermalization in a six-site Bose-Hubbard system, realized with atoms in an optical lattice.In this work, we present and experimentally demonstrate a new protocol to measure the second-order Rényi entropy, S (2) , based on, and extending, the proposals of (19)(20)(21)(22).…”

“…Recently, a protocol to directly measure the second-order Rényi entropy, S (2) , has been demonstrated, requiring collective measurements to be made on two identical copies ρ of a quantum system (15)(16)(17)(18). In (17), that protocol was used to study entanglement growth and thermalization in a six-site Bose-Hubbard system, realized with atoms in an optical lattice.…”