Qubit-efficient entanglement spectroscopy using qubit resets

Abstract

One strategy to fit larger problems on NISQ devices is to exploit a tradeoff between circuit width and circuit depth. Unfortunately, this tradeoff still limits the size of tractable problems since the increased depth is often not realizable before noise dominates. Here, we develop qubit-efficient quantum algorithms for entanglement spectroscopy which avoid this tradeoff. In particular, we develop algorithms for computing the trace of the n-th power of the density operator of a quantum system, Tr(ρn), (related to the Rényi entropy of order n) that use fewer qubits than any previous efficient algorithm while achieving similar performance in the presence of noise, thus enabling spectroscopy of larger quantum systems on NISQ devices. Our algorithms, which require a number of qubits independent of n, are variants of previous algorithms with width proportional to n, an asymptotic difference. The crucial ingredient in these new algorithms is the ability to measure and reinitialize subsets of qubits in the course of the computation, allowing us to reuse qubits and increase the circuit depth without suffering the usual noisy consequences. We also introduce the notion of effective circuit depth as a generalization of standard circuit depth suitable for circuits with qubit resets. This tool helps explain the noise-resilience of our qubit-efficient algorithms and should aid in designing future algorithms. We perform numerical simulations to compare our algorithms to the original variants and show they perform similarly when subjected to noise. Additionally, we experimentally implement one of our qubit-efficient algorithms on the Honeywell System Model H0, estimating Tr(ρn) for larger n than possible with previous algorithms.

A tensor network discriminator architecture for classification of quantum data on quantum computers

“…In general, tensor networks encode a D dimensional quantum system into a register of qubits on the (D − 1)-dimensional boundary, so this approach is referred to as "holographic [24][25][26][27][28][29]". Holographic approaches have been recently exploited in a trapped-ion quantum computer to measure the entanglement entropy defined by the spectrum of ρhalf−infinite [30,31], as well as for dynamics [32]. In practice, since MPSs of finite bond dimension have a finite correlation length, a finite "burn-in length" N b suffices (set to L in the results that follow), and the boundary state |r can be chosen to facilitate rapid convergence to the steady state (see Appendix A for further algorithmic details).…”

“…The state of the ancilla qubits in a partially contracted network also carries information about the state, but scales only with the boundary of the contracted region. This so-called "holographic" encoding of quantum information [24][25][26][27][28] has been utilized in recent works to measure static and dynamic properties of quantum states on quantum computers [29][30][31][32]. The ability of tensor networks to partially localize information about a quantum state into a reduced-dimensional representation may also bode well for trainability in machine learning applications by avoiding the phenomenon of barren plateaus [33].…”

Ancilla-free continuous-variable SWAP test

“…To the best of our knowledge, a short-depth (linear in L) quantum circuit for diagonalizing the cyclic permutation operator PERM is not known for qubit systems. As a result, the two-copy test has been used for estimating (trρ L A ) 2 from 2L copies of a purification |ψ AB of ρ A [7,8]. That algorithm can also be used in the CV case using results of Section IV B.…”

“…The qudit SWAP gate for general Hilbert space dimension d can be implemented with between three and six two-qudit gates, depending on the set of controlled arithmetic operations that are allowed to supplement the discrete Heisenberg-Weyl gate set {X(x), Z(z) : (x, z) ∈ Z ×2 d }, X(x) |j = |j + x mod d , Z(z) |j = e 2πijz d |j [3,4]. Further applications of the SWAP test and its multi-register generalizations include variational quantum algorithms for estimation of rank, quantum entropies, and quantum Fisher information [5], entanglement spectroscopy and estimation of polynomial functions of quantum states [6][7][8][9], virtual cooling and error mitigation [10,11], implementing nonlinear transformations of quantum states [12].…”

Symmetry protected entanglement in random mixed states

“…In this section, we discuss a way in which the total charge in subsystem B can be measured without totally collapsing the state of B to a pure state. The output state can then be used to calculate the symmetry resolved Rényi entropy or Rényi negativity using the already established protocols in circuit-based quantum computer [106][107][108] or trapped ions [109]. We present the case of the Z 2 symmetry here.…”