Limitations of Quantum Simulation Examined by Simulating a Pairing Hamiltonian Using Nuclear Magnetic Resonance

Abstract

Quantum simulation uses a well-known quantum system to predict the behavior of another quantum system. Certain limitations in this technique arise, however, when applied to specific problems, as we demonstrate with a theoretical and experimental study of an algorithm to find the low-lying spectrum of a Hamiltonian. While the number of elementary quantum gates required does scale polynomially with the size of the system, it increases inversely to the desired error bound ǫ. Making such simulations robust to decoherence using fault-tolerance constructs requires an additional factor of ∼ 1/ǫ gates. These constraints are illustrated by using a three qubit nuclear magnetic resonance system to simulate a pairing Hamiltonian, following the algorithm proposed by Wu, Byrd, and Lidar [1].The unknown properties and dynamics of a given quantum system can often be studied by using a well-known and controllable quantum system to mimic the behavior of the original system. This technique of quantum simulation is one of the fundamental motivations for the study of quantum computation [2][3][4], and is particularly of interest because a quantum simulation may be performed using space and time resources comparable to the original system. Such "efficient" scaling is dramatically better than the exponentially large resource requirements to simulate any general quantum system with a classical computer, as Feynman originally observed [2].Recent work has continued to arouse great interest in quantum simulation, because it offers the possibility of solving computationally hard problems without requiring the resources necessary for algorithms such as factoring [5] and searching [6]. Experimental results have demonstrated simulations of a truncated oscillator and of a three-body interaction Hamiltonian, using a nuclear magnetic resonance (NMR) quantum computer [7,8], and explored various solid-state models on two qubit systems [9][10][11][12]. Interest has also extended to simulating complex condensed matter systems with quantum optical systems [13], demonstrated vividly by the observation of a superfluid to Mott insulator transition in a Bose-Einstein condensate [14].Often overlooked in the discussion of quantum simulations, however, is the question of desired precision (or error ǫ) in the final measurement results. Current quantum simulation techniques generally scale poorly with desired precision; they demand an amount of space or time which increases as 1/ǫ, broadly translating into a number of quantum gates which grows exponentially with the desired number of bits in the final answer. Why is this scaling behavior so poor, and what is its physical origin?Consider as a specific example the problem of calculating the energy gap ∆ between the ground state |G and the first excited state |E 1 , of a Hamiltonian H. ∆ can be found using the following steps: 1) map the Hilbert space of the system to be simulated to n qubits, 2) prepare the computer in the state |Ψ I = c G |G + c E |E 1 , 3) evolve under the Hamiltonian for times t i , 4) extra...

Low depth quantum circuits for Ising models

“…This mapping could provide new ways to perform quantum simulation, via either quantum or classical algorithms for estimating Ising model partition functions. Given some of the difficulties that beset fault tolerant implementations of quantum simulations [45,46] new approaches are certainly desirable.…”

Self-protected quantum algorithms based on quantum state tomography

“…As in our current method, these other methods scale polynomially in N , and this is commonly considered an exponential speedup over the currently known best classical algorithms for the same task. However, for error ǫ (defined above) the number of digits of precision l in the result is l˜log(1/ǫ), and both the scattering circuit and the adiabatic methods require poly(1/ǫ) elementary steps to obtain this precision, due to the use of the (quantum) Fourier transform at the measurement [42]. In contrast, an efficient algorithm would only require poly(log(1/ǫ)) number of steps.…”

“…While our QST based method does not employ a Fourier transform at the measurement, the general arguments used in [41] apply to QST as well, so that our present algorithm does not improve on the precision issue. As discussed in [42], the origin of the poly(1/ǫ) number of steps is in the use of the Trotter formula for the simulation of U . Use of the Solovay-Kitaev theorem [which improves the Trotter poly(1/ǫ) scaling to O(log 2 (1/ǫ)) scaling] does not help when a fault tolerant implementation is considered, since the latter once again leads to the poly(1/ǫ) scaling [42].…”

“…As discussed in [42], the origin of the poly(1/ǫ) number of steps is in the use of the Trotter formula for the simulation of U . Use of the Solovay-Kitaev theorem [which improves the Trotter poly(1/ǫ) scaling to O(log 2 (1/ǫ)) scaling] does not help when a fault tolerant implementation is considered, since the latter once again leads to the poly(1/ǫ) scaling [42]. However, it is important to note that these general bounds do not preclude specific Hamiltonians from being efficiently simulatable in terms of precision requirements; indeed the exponential precision slow-down is avoided in Shor's algorithm due to the manner in which modular exponentiation is carried out [3].…”

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