Reconciling Equations from the Randomized Compiling (RC) and Cycle Error Reconstuction (CER) Papers

Diamond Distance

\begin{equation} \epsilon(\mathcal{E}) = \frac{1}{2}||\mathcal{E}- \mathcal{I}||_\diamond = \sup_\psi ||(\mathcal{E}\otimes \mathcal{I}_d) - \mathcal{I}_{d^2}||_1 \end{equation}

Diamond Distance Bounds

Let $r = 1-F(\mathcal{E})$ be the process infidelity of a map $\mathcal{E}$ with respect to the identity.

\begin{equation} r(\mathcal{E}) \leq \epsilon(\mathcal{E}) \leq d\sqrt{r(\mathcal{E})} \end{equation}

Threefold Product of Correction/Easy/Randomizing Cycle

We will refer as the noisy representation of the threefold product as

\begin{equation} \nu(E’_i) = \nu(T_i^cE_iT_{i-1}) \end{equation}

where $T_i^c:= H_i^\dagger T_i^\dagger H_i$.

Comparing Gate Dependent/Independent noise

Let $\Lambda(E_i’) = \nu(E’_i)\phi(E_i’)^\dagger$ be the noise associated with the cycle $\nu(E’_i) = \nu(T_i^cE_iT_{i-1})$ which can be gate dependent. We will compare this noise model with its correspondent gate independent noise defined as follows: \begin{equation}\label{avg_noise} \Lambda_i^\textbf{T} = \langle \Lambda(E’_i) \rangle_{E’_i} = \langle \Lambda(T_i^cE_iT_{i-1}) \rangle_{T_{i-1}} \end{equation}

Effective Dressed Cycle

\begin{equation} \nu_{\text{drs.}}^{\text{eff.}}(H_i,E_i):= \left \langle \phi(T_i) \nu(H_i)\nu(E’_i)\phi^\dagger(T_{i-1})\right \rangle_{T_i,T_{i-1}} \end{equation}

I will now show that, even in the case where the easy gates have a gate dependent error model, the effective dressed cycle is equivalent to a cycle in which the noise in between the hard and easy cycles is going to be a Pauli twirled noise where the noise from the easy gates is going to be given by Equation \ref{avg_noise}, which is a gate-independent.

\[\begin{aligned} \nu_{\text{drs.}}^{\text{eff.}}(H_i,E_i) &:= \left\langle \phi(T_i)\,\nu(H_i)\,\nu(E'_i)\,\phi^\dagger(T_{i-1}) \right\rangle_{T_i,T_{i-1}}\\ &= \left\langle \phi(T_i)\,\phi(H_i)\,\Lambda(H_i)\,\Lambda(E_i')\,\phi(E'_i)\,\phi^\dagger(T_{i-1}) \right\rangle_{T_i,T_{i-1}}\\ &= \left\langle \phi(H_i)\,\phi^\dagger(T_i^c)\,\Lambda(H_i)\,\Lambda(E_i')\,\phi(E'_i)\,\phi^\dagger(T_{i-1}) \right\rangle_{T_i,T_{i-1}}\\ &= \left\langle \phi(H_i)\,\phi^\dagger(T_i^c)\,\Lambda(H_i)\,\Lambda(T_i^c E_i T_{i-1})\,\phi(T_i^c E_i T_{i-1})\,\phi^\dagger(T_{i-1}) \right\rangle_{T_i,T_{i-1}}\\ &= \phi(H_i)\,\left\langle \phi^\dagger(T_i^c)\,\Lambda(H_i)\,\Lambda(T_i^c E_i T_{i-1})\,\phi(T_i^c) \right\rangle_{T_i,T_{i-1}}\,\phi(E_i)\\ &= \phi(H_i)\,\left\langle \phi^\dagger(T_i^c)\,\Lambda(H_i)\, \left\langle \Lambda(T_i^c E_i T_{i-1}) \right\rangle_{T_{i-1}}\,\phi(T_i^c) \right\rangle_{T_i}\,\phi(E_i)\\ &= \phi(H_i)\,\left\langle \phi^\dagger(T_i^c)\,\Lambda(H_i)\, \langle \Lambda(E'_i) \rangle_{E'_i}\,\phi(T_i^c) \right\rangle_{T_i}\,\phi(E_i)\\ &= \phi(H_i)\,\left\langle \phi^\dagger(T_i^c)\,\Lambda(H_i)\, \Lambda_i^\textbf{T}\,\phi(T_i^c) \right\rangle_{T_i}\,\phi(E_i)\\ &= \phi(H_i)\, S_i\, \phi(E_i) \end{aligned}\]

where $S_i$ is a stochastic Pauli channel.

A circuit with gate-dependent noise on the easy gates in the RC paper is defined as $\mathcal{C}_\text{GD}$. In the CER paper, a circuit with any arbitrary noise model is described as $\left \langle \mathcal{C}_\text{RC}(\vec{T})\right \rangle_{\vec{T}}$. Therefore, in our case, we have that $\mathcal{C}_\text{GD} \equiv \left \langle \mathcal{C}_\text{RC}(\vec{T})\right \rangle_\vec{T}$.

A circuit with gate-independent noise is defined in the RC paper as $\mathcal{C}_\text{GI}$ and its gate-independent error model on the easy gates is given by equation \ref{avg_noise}. Since the noise on the easy gates is gate-independent, then from the proof of Lemma 1 of the CER paper, this circuit exactly factorizes as the product of effective dressed cycles; that is,

\[\begin{aligned} \mathcal{C}_\text{GI} &= \nu_{m:0}^\text{eff} \\ &= \nu_{\text{drs}.}^\text{eff.}(E_m)\nu_{\text{drs}.}^\text{eff.}(H_{m-1},E_{m-1})\ldots \nu_{\text{drs}.}^\text{eff.}(H_0,E_0) \end{aligned}\]

And recall that,

\[\begin{aligned} \left\langle \mathcal{C}_{\mathrm{RC}}(\vec{T}) \right\rangle_{\vec{T}} &= \left\langle \nu_{m:0}^{\mathrm{adj.}} \right\rangle \\ &= \nu_{m:0}^{\mathrm{eff.}} + \sum_{i=1}^{m} \nu_{m:i+1}^{\mathrm{adj.}} \left\langle \delta_i \delta_{i-1} \nu_{i-2:0}^{\mathrm{adj.}} \right\rangle \end{aligned}\]

Therefore,

\[\begin{aligned} \mathcal{C}_\text{GD} - \mathcal{C}_\text{GI} &= \left\langle \mathcal{C}_{\mathrm{RC}}(\vec{T}) \right\rangle_{\vec{T}} - \nu_{m:0}^{\mathrm{eff.}} \\ &= \sum_{i=1}^{m} \nu_{m:i+1}^{\mathrm{adj.}} \left\langle \delta_i \delta_{i-1} \nu_{i-2:0}^{\mathrm{adj.}} \right\rangle \end{aligned}\]

and from Theorem 2 of the RC paper

\[\begin{aligned} \left\|\mathcal{C}_\text{GD} - \mathcal{C}_\text{GI} \right\|_\diamond &= \left\| \sum_{i=1}^{m} \nu_{m:i+1}^{\mathrm{adj.}} \left\langle \delta_i \delta_{i-1} \nu_{i-2:0}^{\mathrm{adj.}} \right\rangle \right\|_\diamond \\ &\leq \underbrace{\sum_{i=1}^{m} \left \langle \left\| \Lambda(E'_i) - \Lambda_i^\textbf{T}\right\|_\diamond \right \rangle_{T_1,\ldots, T_{m-1}}}_{\text{from the RC paper}} \end{aligned}\]

We can also go the other route and apply the triangle inequality.

\[\begin{aligned} \left\|\mathcal{C}_\text{GD} - \mathcal{C}_\text{GI} \right\|_\diamond &= \left\| \sum_{i=1}^{m} \nu_{m:i+1}^{\mathrm{adj.}} \left\langle \delta_i \delta_{i-1} \nu_{i-2:0}^{\mathrm{adj.}} \right\rangle \right\|_\diamond \\ &\leq \underbrace{\sum_{i=1}^{m}\left\| \nu_{m:i+1}^{\mathrm{adj.}} \left\langle \delta_i \delta_{i-1} \nu_{i-2:0}^{\mathrm{adj.}} \right\rangle \right\|_\diamond}_{\text{from the triangle inequality}} \end{aligned}\]

So essentially, I have translated Theorem 2 from the RC paper in terms of the definitions from the CER paper. Clearly there is an upper bound in the perturbation terms of the CER paper; the higher the variance in the error rates of the easy gates the looser the bound is. But the only conclusion is that more variance $\implies$ looser upper bound.