Effects of noise, correlations, and errors in the preparation of initial states in quantum simulations

In principle, a quantum system could be used to simulate another quantum system. The purpose of such a simulation would be to obtain information about problems which are difficult to simulate on a classical computer due to the exponential increase of the Hilbert space with the size of the system and which cannot be readily measured or controlled in an experiment. The system will interact with the surrounding environment and with the other particles in the system, and be implemented using imperfect controls, making it subject to noise. It has been suggested that noise does not need to be controlled to the same extent as it must be for general quantum computing. However, the effects of noise in quantum simulations are not well understood and how best to treat them in most cases is not known. In this paper we study an existing quantum algorithm for simulating the one-dimensional Fano-Anderson model using a liquid-state NMR device. We examine models of noise in the evolution using different initial states in the original model. We also add interacting spins to simulate a realistic situation where an environment of spins is present. We find that states which are entangled with their environment, and sometimes correlated but not necessarily entangled, have an evolution which is described by maps which are not completely positive. We discuss the conditions for this to occur and also the implications.

Figure 1

Schematic representation of the simulated system. Qubit 1 is used to simulate the resonant impurity and qubit 2 represents a fermion site. The two particles interact via the potential $V$.

Figure 2

Schematic representation of the simulated system. Qubit 1 is used to simulate the resonant impurity and Qubit 2 represents a fermion site. The two particles interact via the potential $V$. Qubit 2 interacts with two external spins (qubits 3 and 4) through the coupling term $J_{zz}$.

Figure 3

Schematic representation of the simulated system. Qubit 1 is used to simulate the resonant impurity and qubit 2 represents a fermion site. The two particles interact via the potential $V$, and with an external spin (qubit 3) through the coupling term $J_{zz}$.

Figure 4

Evolution of the Bloch vector of the reduced dynamics of qubit 1 in the initial state ${\rho }_1=|0\rangle \langle 0|$ as a function of time.

Figure 5

Eigenvalues of the dynamical map $B$ of the reduced dynamics of qubit 1. The initial state of the closed system is $|\psi \rangle =|01\rangle$. This is an example of an initially pure state (case A.1) with zero quantum discord. The parameters of the Hamiltonian are $\epsilon =-8$ meV, $\varepsilon =-2$ meV, $V=4$ meV. The evolution was carried out for the time interval $\Delta t\in [0.1,0.9]$. There are four sets of eigenvalues, but due to the form of the dynamical map, two of these sets appear to overlap with the other two sets, which is the reason why only two lines show on the graph.

Figure 6

Eigenvalues of the dynamical map of the reduced dynamics of qubit 1. In the open system two qubits are interacting via $zz$ coupling with qubit 2 with coupling constant $J_{zz}=8$ meV. The initial state of the system and bath is given by $\psi =|0111\rangle$ (case A.4). The system parameters are $\epsilon =-8$ meV, $\varepsilon =-2$ meV, and $V=4$ meV. The evolution is carried out for the time interval $t\in [0.1,0.9]$. The dynamical map of the reduced dynamics for this configuration is also completely positive. Similarly to the case of Fig. 5, there are two sets of eigenvalues which overlap.

Figure 7

Eigenvalues of the dynamical map of the reduced dynamics of qubit 1. The system is open. An additional qubit is interacting via $zz$ coupling with qubits 1 and 2 with coupling constant $J_{zz}=1/10$ meV. The initial state of the system and bath is $|\psi \rangle =|011\rangle$ (case A.4 with only one additional qubit). The system parameters are $\epsilon =-8$ meV, $\varepsilon =-2$ meV, and $V=4$ meV. The evolution is carried out for the time interval $t\in [0.1,0.9]$. This configuration was the same as in Fig. 5, because the couplings to the third qubit both had the same magnitude, which results in a shift in the values of the energies, but the relative sizes of the parameters remain unchanged.

Figure 8

Time correlation function of the reduced dynamics of qubit 1. Qubit 2 is interacting via $zz$ coupling with qubits 3 and 4. The coupling strengths are $J_{zz}=1$ meV in the “weak” regime, $J_{zz}=6$ meV in the “intermediate” regime, and $J_{zz}=9$ meV in the “strong” regime. The initial state of the system and bath is $|\psi \rangle =|0111\rangle$ (case A.4). The system parameters are $\epsilon =-8$ meV, $\varepsilon =-2$ meV, and $V=4$ meV, for times $t\in [0.1,0.9]$.

Figure 9

Animation of the evolution of the Bloch vector of the reduced dynamics of qubit 1 in the initial state ${\rho }_1=\frac{1}{2}(1\mathrm{l}+0.2{\sigma }_x+0.97{\sigma }_z)$.

Figure 10

Eigenvalues of the dynamical map of the reduced dynamics of qubit 1. The initial states of the qubits in the closed system are ${\rho }_1=\frac{1}{2}(1\mathrm{l}+0.2{\sigma }_x+0.97{\sigma }_z)$ and ${\rho }_2=\frac{1}{2}(1\mathrm{l}-{\sigma }_z)$ (case B.1). The parameters of the Hamiltonian are $\epsilon =-8$ meV, $\varepsilon =-2$ meV, $V=4$ meV. The evolution is carried out for the time interval $t\in [0.1,0.9]$.

Figure 11

Eigenvalues of the dynamical map for the reduced dynamics of qubit 1. The initial states of the particles in the system are ${\rho }_1=\frac{1}{2}(1\mathrm{l}+0.2{\sigma }_x+0.97{\sigma }_z)$ and ${\rho }_2=\frac{1}{2}(1\mathrm{l}-{\sigma }_z)$. The initial states of the particles that compose the spin bath are ${\rho }_3=\frac{1}{2}(1\mathrm{l}-{\sigma }_z)$ and ${\rho }_4=\frac{1}{2}(1\mathrm{l}-{\sigma }_z)$. The total state of the bath is an example of case B.3. The Hamiltonian parameters are $\epsilon =-8$ meV, $\varepsilon =-2$ meV, $V=4$ meV. The coupling to the bath has strength $J_{zz}=6$ meV. The evolution is carried out in the time interval $t\in [0.1,0.9]$.

Figure 12

Eigenvalues of the dynamical map for the reduced dynamics of qubit 1. The initial states of the particles in the system are ${\rho }_1=\frac{1}{2}(1\mathrm{l}+0.2{\sigma }_x+0.97{\sigma }_z)$ and ${\rho }_2=\frac{1}{2}(1\mathrm{l}-{\sigma }_z)$. The initial state of the additional qubit is ${\rho }_3=\frac{1}{2}(1\mathrm{l}-{\sigma }_z)$. This is another form of case B.3, except that there is only one additional qubit acting as the bath. The Hamiltonian parameters are $\epsilon =-8$ meV, $\varepsilon =-2$ meV, $V=4$ meV. The coupling to the additional qubit has strength $J_{zz}=\frac{1}{10}$ meV. The evolution of the system is evaluated for the time interval $t\in [0.1,0.9]$.

Figure 13

Time correlation function of the reduced dynamics of qubit 1. The Hamiltonian parameters are $\epsilon =-8$ meV, $\varepsilon =-2$ meV, $V=4$ meV, for time interval $t\in [0.1,1]$. These results represent the evolution of the closed system, the system where qubit 2 interacts with two additional qubits, the system in which an additional qubit that interacts with qubits 1 and 2. This was done when qubit 1 was in the initial states $\rho =|0\rangle \langle 0|$ and $\rho =\frac{1}{2}(1\mathrm{l}+0.2{\sigma }_x+0.97{\sigma }_z)$, as indicated above.

Figure 14

Time correlation function of the reduced dynamics of qubit 1. The system parameters are $\epsilon =-8$ meV, $\varepsilon =-2$ meV, $V=4$ meV in the time interval $t\in [0.1,0.9]$. The results correspond to the closed system and the system that interacts with two additional qubits, coupled only to qubit 2. The initial state of qubit 1 is $\rho =|0\rangle \langle 0|$ for one set of results, and $\rho =\frac{1}{2}(1\mathrm{l}+0.2{\sigma }_x+0.97{\sigma }_z)$ for the other.

Figure 15

Time correlation function of the reduced dynamics of qubit 1. The system parameters are $\epsilon =-8$ meV, $\varepsilon =-2$ meV, $V=4$ meV evaluated in the time interval $t\in [0.1,0.9]$. The result represents the time correlation function of the closed system compared to the correlation function of the reduced dynamics of qubit 1 in the initial state $\rho =\frac{1}{2}(1\mathrm{l}+a_1{\sigma }_x+a_3{\sigma }_z)$ for different values of $a_1$ and $a_3$.

Figure 16

Time correlation function of the reduced dynamics of qubit 1. The system parameters are $\epsilon =-8$ meV, $\varepsilon =-2$ meV, $V=4$ meV evaluated in the time interval $t\in [0.1,0.9]$. The results for the time correlation function of the reduced dynamics of qubit 1 for: a closed system where the two qubits are in a pure state (case A.1 label $\times$); a system where the initial state is a correlated one with ${\rho }_1^I=\frac{1}{2}(1\mathrm{l}+0.2{\sigma }_x+0.97{\sigma }_z)$, ${\rho }_1^{II}=\frac{1}{2}(1\mathrm{l}+{\sigma }_z)$, ${\rho }_2^I={\rho }_2^{II}=\frac{1}{2}(1\mathrm{l}-{\sigma }_z)$, and $p=\frac{1}{2}$ (case B.2 label $-$); a system where the initial state is a correlated one with ${\rho }_1^I=\frac{1}{2}(1\mathrm{l}+0.2{\sigma }_x+0.97{\sigma }_z)$, ${\rho }_1^{II}=\frac{1}{2}(1\mathrm{l}-0.2{\sigma }_x-0.97{\sigma }_z)$, ${\rho }_2^I=\frac{1}{2}(1\mathrm{l}-{\sigma }_z)$, ${\rho }_2^{II}=\frac{1}{2}(1\mathrm{l}+{\sigma }_z)$, and $p=\frac{1}{2}$ (case B.2 label $△$); a system where the initial state is a correlated one (case A.3) where ${\rho }_1^I=\frac{1}{2}(1\mathrm{l}+{\sigma }_z)$, ${\rho }_1^{II}=\frac{1}{2}(1\mathrm{l}-{\sigma }_z)$, ${\rho }_2^I=\frac{1}{2}(1\mathrm{l}-{\sigma }_z)$, ${\rho }_2^{II}=\frac{1}{2}(1\mathrm{l}+{\sigma }_z)$, and $p=\frac{1}{9}$ ($◯$); a system where the initial state is a maximally entangled one (case A.5) and qubit 2 is interacting with two additional qubits with coupling strength $J_{zz}=\frac{1}{10}$ meV ($\diamond$).