Robustness of optimal probabilistic storage and retrieval of unitary channels to noise

Quantum information theory and quantum computers themselves, started to draw more attention since the discovery of the first algorithms with potential quantum speed-ups such as factorization or quantum search shor ; grover , or seminal work on quantum cryptography QKD . The theory itself is bounded by no-go theorems noclon ; nobroadcast ; nohiding ; nodelete - theorems that put limitations on what the quantum processing of information can achieve. There exist two different conceptual ways how to circumvent the no-go theorems: either give up the exactness, or the determinism of the performance. In other words one is either satisfied only with approximate solutions or one keeps the requirement of exact solutions, with the caveat of only a certain probability of successfully reaching them.

At the heart of a classical computer science lies the concept of processor. Therefore, one envisions a similar device also for a quantum computer - a so-called quantum processor nielsen1 . However, due to the no-programming theorem nielsen1 , one can never construct a truly universal quantum processor capable of executing all transformations with absolute precision. One way of dealing with the no-programming theorem is to define processors that are able to implement the exact transformation however only with some probability probProc1 ; probProc2 (the other way is to use approximate processors appProc3 ; appProc4 ).

In a sense complementary problem to quantum programming is known as quantum learning. The aim is to design protocols to store and retrieve the action of an unknown quantum transformations. Informally, the goal of the storage phase is to exploit the transformation device given number of times $N$ and imprint its transformation into a quantum state of some memory system in a way that during the retrieval phase this state enables us (using the quantum processor) to implement the stored transformation given number of times $M$. There are several variations of this problem and few of them has been already considered quantLearning ; storingQuantDyn ; STORunitary ; STORPhaseGate . In this work, we investigate the noise robustness of optimal probabilistic storage and retrieval device (PSAR) for qubit phase gates introduced in STORPhaseGate .

Following STORPhaseGate consider a one-parametric set of unitary gates $U_{\varphi}=|0\rangle\langle 0|+e^{i\varphi}|1\rangle\langle 1|$ acting on a qubit Hilbert space $\mathcal{H}_{2}$. We assume that during the storing phase (see Figure 1)we can access the unknown unitary gate $N$ times and produce a state $\omega_{U}$ (memory). This state is used in the retrieval phase to restore the action of the gate with a probability of success $p_{\rm success}=N/(N+1)$ independent of $U_{\varphi}$.

In this work we aim to analyze the performance of the PSAR devices introduced in STORPhaseGate in cases when during the storing phase the ideal unitary gates are replaced by their noisy versions. In particular, we consider two different noise models: depolarizing and phase-damping ones. The retrieved channels will be compared with the stored noisy gates, but also with the noiseless unitary gates. The paper is organized as follows: in Section II we describe the mathematical framework of quantum networks used to formulate and solve the problem. Section III provides mathematical formulation of the problem and introduces the noise models we investigate. In Section IV we discuss in details the case of $2\to 1$ PSAR and Section V contains generalizations for $N\to 1$ case. The implementations are studied in Section VI and Section VII summarized the conclusions of the performed analysis.

Mathematical formalism used in this paper stems from the framework of quantum networks QN ; higherOrderQT ; architecture ; QN2 . Let us denote by ${\cal{L}}(\mathcal{H}_{a})\equiv{\cal{L}}_{a}$ the set of linear operators on a finite-dimensional Hilbert space $\mathcal{H}_{a}$ and by ${\cal{L}}(\mathcal{H}_{a},\mathcal{H}_{b})\equiv{\cal{L}}_{ab}$ the set of linear operators from $\mathcal{H}_{a}$ to $\mathcal{H}_{b}$. Quantum operation ${\cal{O}}:{\cal{L}}_{a}\to{\cal{L}}_{b}$ is a completely positive trace non-increasing linear map. Trace-preserving quantum operation, thus mapping quantum states into quantum states, is called quantum channel.

There exists a one-to-one correspondence (Choi-Jamiolkowski isomorphism) between linear operators $A\in{\cal{L}}(\mathcal{H}_{a},\mathcal{H}_{b})$ and vectors $|A\rangle\!\rangle\in\mathcal{H}_{a}\otimes\mathcal{H}_{b}$:

where $A_{mn}=\langle m|A|n\rangle$, $\{|m\rangle\}$ and $\{|n\rangle\}$ are orthonormal basis of $\mathcal{H}_{a}$ and $\mathcal{H}_{b}$, respectively. Similarly, every quantum operation ${\cal{O}}$ is associated with a positive operator (Choi operator)

where ${\cal{I}}_{a}$ is an identity map on ${\cal{L}}(\mathcal{H}_{a})$ and $I\in{\cal{L}}_{aa}$ is the identity operator, i.e. $|I\rangle\!\rangle=\sum_{n}|n\rangle\otimes|n\rangle$.

Consider a composition of two operations ${\cal{O}}:{\cal{L}}_{a}\to{\cal{L}}_{b})$ and $\widetilde{\cal{O}}:{\cal{L}}_{b}\to{\cal{L}}_{c}$. Then, the Choi operator of the composition $\widetilde{\cal{O}}\circ\cal{O}$ is expressed via link product denoted by $\star$ and defined as follows:

where $T_{b}$ denotes partial transposition on space $\mathcal{H}_{b}$ and $\mathbb{I}_{a}$ stands for the identity operator on $\mathcal{H}_{a}$.

Quantum network ${\cal{R}}$ (also known as quantum comb) is a concatenation of quantum operations ${\cal{O}}_{1},\cdots,{\cal{O}}_{N}$ (see Fig. (2)), where some outputs of the preceding operation are connected with some inputs of the subsequent one. The connectivity structure is included in the definition of each particular link product in the whole sequence of compositions. As a result after linking operations some of the outputs and inputs remain open, thus, allowing the quantum network to accept quantum channels and operations of a suitable input-output type at its inputs. Overall the quantum network ${\cal{R}}$ transforms $N-1$ operations into one quantum operation. The associated Choi operator of quantum network ${\cal{R}}$ is an operator $R^{N}=O_{1}\star O_{2}\star\cdots\star O_{N}$ acting on the Hilbert space ${\cal{H}_{\rm in}}\otimes{\cal{H}_{\rm out}}$, where ${\cal{H}}_{\rm in}=\otimes_{i=0}^{N-1}{\cal{H}}_{2i}$ represents the causally ordered input spaces and ${\cal{H}}_{\rm out}=\otimes_{i=0}^{N-1}{\cal{H}}_{2i+1}$ stands for the output spaces. Choi operators of deterministic (trace-preserving) quantum networks obey the recursive normalization conditions:

for $k=1,\cdots,N$. For probabilistic quantum network ${\cal{S}}$ there always exists a deterministic quantum network ${\cal{R}}$ such that $S\leq R$. Collection of probabilistic quantum networks ${\cal{S}}_{1},\cdots,{\cal{S}}_{m}$ summing up to a deterministic quantum network $\sum_{x}{\cal{S}}_{x}={\cal{R}}$ form a generalized quantum instrument. Generalized quantum instrument fulfills a similar role for quantum networks as does quantum instrument for quantum channels. The index $x$ represents the corresponding classical outcome.

The optimal probabilistic storage and retrieval (PSAR) for phase gates introduced in STORPhaseGate consists of the probe state $|\Psi\rangle_{AA^{\prime}}$, where $A$ labels all inputs of $N$ phase gates and $A^{\prime}$ represent the ancillary part of the memory, and the retrieval operation ${\cal R}_{s}$ using the memory $M$ to implement the phase gate action on the unknown input system $C$ that is transformed into the system $D$ (see Fig. 1).

In particular, during the storing this state is transformed into

where $U_{\varphi}=|0\rangle\langle 0|+e^{i\varphi}|1\rangle\langle 1|$ is the phase gate. It was derived in STORPhaseGate that optimal probe state takes the form

where $\Pi_{j}$ are projectors onto Hilbert subspaces ${\cal H}_{j}$. These subpsaces follows from the irreducible representations of $U(1)$

where $I_{m_{j}}$ stands for identity operator in the multiplicity subspace ${\cal H}_{m_{j}}$, thus, inducing the decomposition ${\cal H}_{A}=\sum_{j}{\cal H}_{j}\otimes{\cal H}_{m_{j}}$ with one-dimensional ${\cal H}_{j}$. Consequently,

We may set $\Pi_{j}=|\overline{j}\rangle\langle\overline{j}|$ with $|\overline{j}\rangle=|0^{\otimes(N-j)}\rangle\otimes|1^{\otimes j}\rangle\in{\cal H}_{2}^{\otimes N}$. Then $|\Psi\rangle_{AA^{\prime}}=(1/\sqrt{N+1})\sum_{j}|\overline{j}\overline{j}\rangle$.

Let us denote the memory state as $\Psi_{\varphi}=|\Psi_{\varphi}\rangle\langle\Psi_{\varphi}|$. It was shown in STORPhaseGate that using the Choi operator

associated with the retrieval operation ${\cal R}_{s}$ the action of the retrieval is evaluated as follows

where $|\Psi_{\varphi}^{*}\rangle=(1/\sqrt{N+1})\oplus_{j}e^{-ij\varphi}|\Pi_{j}\rangle\!\rangle$ and $N/(N+1)$ is the success probability, because

There are several variations how the noise can enter the design of the storing and retrieval procedures. In what follows we will assume the performance of the black box PSAR introduced in STORPhaseGate is unchanged and only the phase gates we are aiming to store are noisy. That is, instead of ${\cal U}_{\varphi}(\cdot)=U_{\varphi}\cdot U_{\varphi}^{\dagger}$ the ”phase gates” implement the following transformation ${\cal E}_{\varphi}=q{\cal U}_{\varphi}+(1-q){\cal N}$, where ${\cal N}$ is the noise. We will consider two types of noise:

• •

  Depolarisation. In this case we set ${\cal N}={\cal C}_{I/2}$, where ${\cal C}_{I/2}$ stands for the completely depolarizing noise (also known as white noise) transforming any quantum state $\varrho$ into the complete mixture, i.e. ${\cal{C}}_{I/2}(\varrho)=I/2$. The phase gate implements the channel

  $\displaystyle{\cal{E}}_{\varphi}=q{\cal{U}}_{\varphi}+(1-q){\cal{C}}_{I/2},$

  being a convex combination ($q\in[0,1]$) of the desired unitary channel and a completely depolarizing noise ${\cal C}_{I/2}$ with the Choi operator $C=\frac{1}{2}(|00\rangle\langle 00|+|01\rangle\langle 01|+|10\rangle\langle 10|+|11\rangle\langle 11|)=(I\otimes I)/2$.

• •

  Dephasing. In this case we set ${\cal N}={\cal P}$, where ${\cal P}$ stands for the completely dephasing channel for all states $\varrho$ diminishing off-diagonal terms in the computational basis identified with eigenvectors of $\sigma_{z}$ operator, i.e. ${\cal P}(\varrho)=\frac{1}{2}(\varrho+\sigma_{z}\varrho\sigma_{z})={\rm diag}[\varrho]=\langle 0|\varrho|0\rangle|0\rangle\langle 0|+\langle 1|\varrho|1\rangle|1\rangle\langle 1|$. That is, phase gates are implementing convex combination

  $\displaystyle{\cal{F}}_{\varphi}=q{\cal{U}}_{\varphi}+(1-q){\cal{P}},$

  where $q\in[0,1]$ and the Choi operator of the completely dephasing channel reads $P=|00\rangle\langle 00|+|11\rangle\langle 11|$.

In this section we will investigate in details the situation when the unknown noisy phase gate device is used twice in the storing phase. The situation is depicted in Fig. 3. For brevity we introduce the following notation for the involved Hilbert spaces ${\cal{H}}_{13}={\cal{H}}_{A}$, ${\cal{H}}_{24}={\cal{H}}_{B}$, ${\cal{H}}_{0}={\cal{H}}_{C}$ and ${\cal{H}}_{5}={\cal{H}}_{D}$. Choi operator for two uses of the considered channel is given as follows:

Unitary phase gate part can be written as follows

where we introduced the vectors $|\overline{0}\rangle\equiv|00\rangle$, $|\overline{1}\rangle\equiv|01\rangle$, $|\overline{2}\rangle\equiv|11\rangle$, $|\overline{3}\rangle\equiv|10\rangle$ and $I_{m_{j}}$ denotes the identity operator on multiplicity spaces. Using this notation the probe state $|\Psi\rangle$ reads

Let us start the general $N\to 1$ analysis with the depolarization noise. Performing a direct calculation of the retrieval for PSAR derived in STORPhaseGate we find out that for arbitrary $N$ if we leave $\frac{1}{N+1}$ in front of all the parentheses then the factors next to $|U_{\varphi}\rangle\!\rangle\langle\!\langle U_{\varphi}|$ follow the pattern:

Similarly the factors in front of $|00\rangle\langle 00|+|11\rangle\langle 11|$ are of the form:

In these expressions we can identify binomial distribution and the whole action of the retrieval process results in the retrieved transformation

Using this result we can formulate the following theorem.

We can see that with the increasing number of uses of noisy phase gates, the probability of successful retrieval is vanishing as $N$ goes to infinity, however, the quality of the retrieval is exactly the same.

The same analysis for the case of dephasing noise is more straighforward. The calculation gives that the retrieved channel is independent of the number of uses $N$, thus, the optimal PSAR for phase gates does the same job also for their noisy version if the noise is modeled by the discussed dephasing.

In this part we will study two different implementations of optimal PSAR device for phase gate: first one due to Vidal-Masanes-Cirac from Ref. storingQuantDyn limited to $N=2^{k}-1$ uses and the one reported in Ref. STORPhaseGate minimizing the size of the storage register to single qudit. Both are equivalent if noiseless phase gates are considered, however, in the presence of noise they may lead to different results.