PEET: a Matlab tool for estimating physical gate errors in quantum information processing systems

Abstract

A Physical Error Estimation Tool (PEET) is introduced in Matlab for predicting physical gate errors of quantum information processing (QIP) operations by constructing and then simulating gate sequences for a wide variety of user-defined, Hamiltonian-based physical systems. PEET is designed to accommodate the interdisciplinary needs of quantum computing design by assessing gate performance for users familiar with the underlying physics of QIP, as well as those interested in higher-level computing operations. The structure of PEET separates the bulk of the physical details of a system into Gate objects, while the construction of quantum computing gate operations are contained in GateSequence objects. Gate errors are estimated by Monte Carlo sampling of noisy gate operations. The main utility of PEET, though, is the implementation of QuantumControl methods that act to generate and then test gate sequence and pulse-shaping techniques for QIP performance. This work details the structure of PEET and gives instructive examples for its operation.

Appendix

1.1 Gate and GateSequence properties and methods

See Table 6.

Table 6 Additional methods

1.2 Calculation of the process matrix

Propagation to calculate $\chi _{{\mathcal {E}}}$ is performed by first transforming Eq. (1) into a set of differential equations for the matrix elements of $\rho$ in Liouville space. This restructures the system dynamics into a Hilbert space of dimension $N^2$ and allows for succinct calculation of process matrices through an evolution superoperator ${\mathcal {L}}$. First, vectorizing $\rho$ columnwise to $\mathbf {\rho }$ yields [29],

$$\begin{aligned} \mathbf {\rho } = \left[ \rho _{11}, \rho _{21},\ldots \rho _{N1}, \rho _{12} \ldots \ldots \rho _{NN} \right] ^T. \end{aligned}$$

(41)

By observing the action of operators M on $\rho$, analogous operations transform upon this vectorization to

$$\begin{aligned} M\rho\Rightarrow & {} (I \otimes M) \mathbf {\rho } \end{aligned}$$

(42)

$$\begin{aligned} \rho M\Rightarrow & {} (M^{T}\otimes I) \mathbf {\rho }, \end{aligned}$$

(43)

and the Linblad equation can be reformulated as

$$\begin{aligned} \frac{\partial }{\partial t} \mathbf {\rho }(t)= & {} {\mathcal {L}}(t) \mathbf {\rho }(t) \nonumber \\= & {} \left( -\frac{i}{\hbar }\left[ I \otimes H(t) - H^{T}(t) \otimes I \right] +\right. \nonumber \\&\left. \sum _i^P \frac{1}{T_1} \left[ \sigma _{+}^{*(i)} \otimes \sigma _{+}^{(i)} - \frac{1}{2} I \otimes \sigma _{+}^{\dag (i)}\sigma _{+}^{(i)} - \frac{1}{2} \sigma _{+}^{\dag (i)}\sigma _{+}^{(i)} \otimes I \right] \right. \nonumber \\&\left. \sum _i^M \frac{1}{T_{\phi }}\left[ \sigma _z^{*(i)} \otimes \sigma _z^{(i)} - \frac{1}{2} I \otimes \sigma _z^{\dag (i)}\sigma _z^{(i)} - \frac{1}{2} \sigma _z^{\dag (i)}\sigma _z^{(i)} \otimes I \right] \right) \mathbf {\rho }(t).\nonumber \\ \end{aligned}$$

(44)

The density matrix $\mathbf {\rho }(t)$ is calculated by exponentiation of the time-dependent superoperator ${\mathcal {L}}(t)$ as

$$\begin{aligned} \mathbf {\rho }(t) = {\hat{\mathcal {T}}} \exp \left( \int _0^t {\mathcal {L}}(s) \mathrm{d}s \right) \mathbf {\rho }(0), \end{aligned}$$

(45)

where ${\hat{\mathcal {T}}}$ is the time-ordering operator [43]. Gate operation dynamics are discretized over $N_t$ points on an interval of [0, T] with resolution $\varDelta t$. For reasonably small $\varDelta t$, short-time propagation methods accurately describe propagation of $\mathbf {\rho }$,

$$\begin{aligned} \mathbf {\rho }(t+\varDelta t)= & {} \exp \left( {\mathcal {L}}(t+\varDelta t) \varDelta t \right) \mathbf {\rho }(t)\\ \mathbf {\rho }(T)= & {} \chi _{{\mathcal {E}}} \mathbf {\rho }(0) \nonumber \end{aligned}$$

(46)

$$\begin{aligned}= & {} \varPi _{j=1}^{N_t} \exp \left( {\mathcal {L}}(j\varDelta t) \varDelta t \right) \mathbf {\rho }(0), \end{aligned}$$

(47)

where the initial process matrix propagator is the $N^2 \times N^2$ identity

$$\begin{aligned} \exp \left( {\mathcal {L}}(0) \varDelta t \right) = {\mathbb {I}}_{N^2}. \end{aligned}$$

(48)

The process matrix propagates to yield $\mathbf {\rho }(T)$ , and its final-time form used in J is simply

$$\begin{aligned} \chi _{{\mathcal {E}}} = \varPi _{j=1}^{N_t} \exp \left( {\mathcal {L}}(j \varDelta t) \varDelta t \right) . \end{aligned}$$

(49)

1.3 Noise generation

Generating random instances of noise following a given noise correlation function are generated in PEET with a Karhunen-Loéve (KL) filter [44]. Representing a disturbance $\delta \beta (t)$ over discrete time points $t_i$ for $i = 1,2,... N_t$ , the finite-dimensional noise correlation matrix $R(t_i,t_j)$ is first diagonalized,

$$\begin{aligned} R = M\varLambda M^{\dag }, \end{aligned}$$

(50)

where M is an orthonormal matrix with column vectors being the eigenvectors of R and $\varLambda$ is the diagonal eigenvalue matrix. The correlation matrix of a random time vector $\mathbf {x}$ (i.e., Matlab’s rand routine) drawn from a normal distribution of zero mean and standard deviation equal to 1 is given by a delta function, $R_x(t_i,t_j) = \langle \mathbf {x}(t_i),\mathbf {x}(t_j) \rangle = \delta _{i,j}$, and can be used to construct the discretized noise signal $\beta (t) \rightarrow \mathbf {\delta \beta }(t_i)$ as

$$\begin{aligned} \mathbf {\delta \beta }(t_i) = \sum _{k=1}^{N_t} {\hat{x}}(t_k)\sqrt{\varLambda _{kk}}M_{ki}^{\dag }. \end{aligned}$$

(51)

where $\mathbf {\delta \beta }(t_i)$ possesses the appropriate noise correlation function,

$$\begin{aligned} \langle \mathbf {\delta \beta }(t_i) \mathbf {\delta \beta }(t_j) \rangle= & {} \sum _{l,k} \langle M_{i,l} \sqrt{\varLambda _{ll}} \mathbf{x}(t_l)\mathbf{x}(t_k)\sqrt{\varLambda _{kk}}M_{k,j}^{\dag } \rangle \nonumber \\= & {} \sum _{l,k} M_{i,l}\sqrt{\varLambda _{ll}} \sqrt{\varLambda _{kk}}M_{k,j}^{\dag } \langle \mathbf{x}(t_l),{\mathbf{x}_k}(t_k) \rangle \nonumber \\= & {} \sum _{k} M_{i,k}\varLambda _{kk} M_{k,j}^{\dag } \nonumber \\= & {} R(t_i,t_j). \end{aligned}$$

(52)

The KL filter is a useful for “shot-to-shot” noise that only endures for a single gate. Scaling this method to background noise over an entire GateSequence though, leads to difficulties storing and diagonalizing the correlation matrix. In contrast, for simple noise forms a frequency mask $g(\omega )$ is applied to white noise to generate low-frequency noise, 1 / f noise, etc,

$$\begin{aligned} z(t) = {\mathcal {F}}^{-1}\left[ g(\omega ) {\mathcal {F}}\left[ x(t) \right] \right] , \end{aligned}$$

(53)

where ${\mathcal {F}}[\cdot ]$ is the Fourier transform, as in Eq. (3).

1.4 Hamiltonian types

Below are descriptions of the Hamiltonians for the supported type specifiers, their cont specification, and corresponding noise terms.

type=’l1st’:

• Hamiltonian

  $$\begin{aligned} {\tilde{H}}(t) =\frac{\omega + \delta \omega (t)}{2} Z + (1+\delta c_2(t))[c(t)+\delta c_1(t)] \mu \end{aligned}$$

  (54)

• cont: $1\times N_t$ vector $\rightarrow c(t)$

• system noise: $\delta \omega (t)$

• additive control noise: $\delta c_1(t)$

• multiplicative control noise: $\delta c_2(t)$

type=’l1rt’:

• Hamiltonian

  $$\begin{aligned} {\tilde{H}}(t) = \frac{(1+\delta A_2(t))[A(t)+\delta A_1(t)]}{2}\left( \begin{array}{c c} 0 &{} \quad e^{-\frac{i}{\hbar }\delta \omega (t) t -i \phi }\\ e^{\frac{i}{\hbar } \delta \omega (t) t +i \phi } &{} \quad 0\\ \end{array} \right) . \end{aligned}$$

  (55)

• cont: $1\times N_t+1$ vector $\rightarrow [A(t),\phi ]$

• system noise: $\delta \omega (t)$

• additive control noise: $\delta A_1(t)$

• multiplicative control noise: $\delta A_2(t)$

type=’lqst’:

• Hamiltonian

  $$\begin{aligned} {\tilde{H}}(t) =\sum _i^M\frac{\omega _i+ \delta \omega _i(t)}{2} Z + (1+\delta c_{i,2}(t))[c_i(t)+\delta c_{i,1}(t)] \mu _i \end{aligned}$$

  (56)

• cont: $1\times N_t*M$ vector $\rightarrow [c_1(t),c_2(t),...c_M(t)]$

• system noise: $[\delta \omega _1(t),\delta \omega _2(t),...\delta \omega _M(t)]$

• additive control noise: $[\delta c_{1,1}(t),\delta c_{2,1}(t),...\delta c_{M,1}(t)]$

• multiplicative control noise: $[\delta c_{1,2}(t),\delta c_{2,2}(t),...\delta c_{M,2}(t)]$

type=’ldst’:

• Hamiltonian

  $$\begin{aligned} {\tilde{H}}(t)= & {} \left( \begin{array}{lllc} E_1 + \delta E_1(t) &{} 0 &{} 0 &{} \quad 0\\ 0 &{} E_2 + \delta E_2(t) &{} 0 &{} \quad 0\\ \vdots &{} &{} \ddots &{} \quad \vdots \\ 0 &{} 0 \ldots &{} &{} E_N + \delta E_N(t) \\ \end{array} \right) + \nonumber \\&\quad (1+\delta c_2(t))[c(t)+\delta c_1(t)] \mu \end{aligned}$$

  (57)

• cont: $1\times N_t$ vector $\rightarrow c(t)$

• system noise: $\rightarrow [\delta E_1(t),\delta E_2(t),...\delta E_N(t)]$

• additive control noise: $\delta c_1(t)$

• multiplicative control noise: $\delta c_2(t)$

type=’lgst’:

• Hamiltonian

  $$\begin{aligned} {\tilde{H}}(t)= & {} \left( \begin{array}{lllc} E_1 + \delta E_1(t) &{} 0 &{} 0 &{} \quad 0\\ 0 &{} E_2 + \delta E_2(t) &{} 0 &{} \quad 0\\ \vdots &{} &{} \ddots &{} \quad \vdots \\ 0 &{} 0 \ldots &{} &{}E_N + \delta E_N(t) \\ \end{array} \right) + \nonumber \\&\quad \sum _i^M (1+\delta c_{i,2}(t))[c(t)+\delta c_{i,1}(t)] \mu _i \end{aligned}$$

  (58)

• cont: $1\times N_t*M$ vector $\rightarrow [c_1(t),c_2(t),...c_M(t)]$

• system noise: $\rightarrow [\delta E_1(t),\delta E_2(t),...\delta E_N(t)]$

• additive control noise: $[\delta c_{1,1}(t),\delta c_{2,1}(t),...\delta c_{M,1}(t)]$

• multiplicative control noise: $[\delta c_{1,2}(t),\delta c_{2,2}(t),...\delta c_{M,2}(t)]$

type=’lgsp’:

• Hamiltonian

  $$\begin{aligned} {\tilde{H}}(t)= & {} \left( \begin{array}{lllc} E_1 + \delta E_1(t) &{} 0 &{} 0 &{} \quad 0\\ 0 &{} E_2 + \delta E_2(t) &{} 0 &{} \quad 0\\ \vdots &{} &{} \ddots &{} \quad \vdots \\ 0 &{} 0 \ldots &{} &{} E_N + \delta E_N(t) \\ \end{array} \right) +\nonumber \\&\quad \sum _i^M (1+\delta c_{i,2}(t))[c[\lambda _j](t)+\delta c_{i,1}(t)] \mu _i \end{aligned}$$

  (59)

• cont: $1\times Q$ vector $\rightarrow [\lambda _1, \lambda _2,...\lambda _Q]$

• system noise: $\rightarrow [\delta E_1(t),\delta E_2(t),...\delta E_N(t)]$

• additive control noise: $[\delta c_{1,1}(t),\delta c_{2,1}(t),...\delta c_{M,1}(t)]$

• multiplicative control noise: $[\delta c_{1,2}(t),\delta c_{2,2}(t),...\delta c_{M,2}(t)]$