## 8 Tomography

The main task of tomography is to reconstruct a full classical description of an n-qubit state from experimental data. In the process of developing methods of tomography, the following question entitled shadow tomography problem is raised. Given an n-qubit state $\rho$ and two-outcome measurements $M_1, M_2, ..., M_m$, estimate tr[$M_i\rho$] up to error $\epsilon$, for $i = 1, 2, ..., m$. This problem is solved by designing a quantum procedure with $\tilde{O}(\epsilon^{-4} \cdot \log^4 m \cdot \log n)$ copies of state, where $\tilde{O}$ hides factors which are polynomial in $\log \log m, \log \log n$ and $\log \epsilon^{-1}$ [39].

After this brief introductory note, let us describe the framework of classical shadows. Let $\rho$ be an unknown n-qubit state. First we apply a set of unitary evolutions $\rho \mapsto u\rho u^\dagger$, where $u$ is randomly selected from an ensemble $\mathfrak{U}$ and then we measure the rotated state in the computational basis $\{|b\rangle : b \in \{0, 1\}^n\}$. Here, different ensembles lead to different versions of procedure and in practical scheme, each ensemble unitary should be realizable as an efficient quantum circuit. We suppose that this collection is tomographically complete, i.e. for each $\tau \neq \rho$ there are $u \in \mathfrak{U}$ and $b$ such that $\langle b|u\tau u^\dagger|b\rangle \neq \langle b|u\rho u^\dagger|b\rangle$. The framework of classical shadow estimation is based on the following algorithm.

**Algorithm 3. (Constructing classical shadow)**

1. Prepare a copy of the unknown quantum state $\rho$,

2. Sample $u \sim \mathfrak{U}$ randomly,

3. Transform $u \mapsto u\rho u^\dagger$,

4. Perform a measurement in the computational basis,

5. Apply the inverse of $u$ to the resulting state from 4 for collapsing $\rho$ to $u^\dagger|\hat{b}\rangle\langle\hat{b}|u$, where $\mathbb{P}(\hat{b} = b) = \langle b|u\rho u^\dagger|b\rangle$,

6. Let $M_{\mathfrak{U}}(\rho) := \mathbb{E}(u^\dagger|\hat{b}\rangle\langle\hat{b}|u) = \mathbb{E}_{u \sim \mathfrak{U}} \sum_{b \in \{0,1\}^n} \langle b|u\rho u^\dagger|b\rangle u^\dagger|b\rangle\langle b|u$,

7. Construct a classical snapshot by computing $\hat{\rho} = M_{\mathfrak{U}}^{-1}(u^\dagger|\hat{b}\rangle\langle\hat{b}|u)$.

State $\hat{\rho}$ is called classical shadow, which has a unit trace, but need not be positive semi-definite. We note that tomographically completeness ensures that $M_{\mathfrak{U}}^{-1}$ exists.

Classical shadow is a useful tool in numerical experiments based on random Clifford measurements [40]. We know that a stabilizer circuit can be simulated efficiently in $O(n^2)$ time by applying the algorithm based on tableau representation. This algorithm enables us to simulate systems efficiently with even more than 160 qubits in practice. To test the performance of predicting with classical shadows, we need to implement repeatedly the following efficient protocol.

1. Sample a Clifford unitary $u$ from $C_n$ by applying the algorithm proposed in [41].

2. The action of $u$ on X- and Z-type operators is fully characterized by parameters $(\alpha, \beta, \gamma, \delta, r, t)$ as follows

$$uX_j u^\dagger = (-1)^{r_j} \prod_{i=1}^{n} X_i^{\alpha_{ji}} Z_i^{\beta_{ji}}, \quad uZ_j u^\dagger = (-1)^{t_j} \prod_{i=1}^{n} X_i^{\gamma_{ji}} Z_i^{\delta_{ji}},$$

where $j = 1, 2, ..., n$ and $r_j, t_j, \alpha_{ji}, \beta_{ji}, \gamma_{ji}$ and $\delta_{ji} \in \{0, 1\}$.

3. A parameterized unitary $u$ by $(\alpha, \beta, \gamma, \delta, r, t)$ can be applied on stabilizer states by changing the stabilizer generators and the destabilizer generators.

4. A computational basis measurement can be simulated by the improved algorithm based on tableau representation.
# 8 Tomography

The main task of tomography is to reconstruct a full classical description of an n-qubit state from experimental data. In the process of developing methods of tomography, the following question entitled shadow tomography problem is raised. Given an n-qubit state $\rho$ and two-outcome measurements $M_1, M_2, ..., M_m$, estimate tr[$M_i\rho$] up to error $\epsilon$, for $i = 1, 2, ..., m$. This problem is solved by designing a quantum procedure with $\tilde{O}(\epsilon^{-4} \cdot \log^4 m \cdot \log n)$ copies of state, where $\tilde{O}$ hides factors which are polynomial in $\log \log m, \log \log n$ and $\log \epsilon^{-1}$ [39].

After this brief introductory note, let us describe the framework of classical shadows. Let $\rho$ be an unknown n-qubit state. First we apply a set of unitary evolutions $\rho \mapsto u\rho u^\dagger$, where $u$ is randomly selected from an ensemble $\mathfrak{U}$ and then we measure the rotated state in the computational basis $\{|b\rangle : b \in \{0, 1\}^n\}$. Here, different ensembles lead to different versions of procedure and in practical scheme, each ensemble unitary should be realizable as an efficient quantum circuit. We suppose that this collection is tomographically complete, i.e. for each $\tau \neq \rho$ there are $u \in \mathfrak{U}$ and $b$ such that $\langle b|u\tau u^\dagger|b\rangle \neq \langle b|u\rho u^\dagger|b\rangle$. The framework of classical shadow estimation is based on the following algorithm.

## Algorithm 3. (Constructing classical shadow)

1. Prepare a copy of the unknown quantum state $\rho$,

2. Sample $u \sim \mathfrak{U}$ randomly,

3. Transform $u \mapsto u\rho u^\dagger$,

4. Perform a measurement in the computational basis,

5. Apply the inverse of $u$ to the resulting state from 4 for collapsing $\rho$ to $u^\dagger|\hat{b}\rangle\langle\hat{b}|u$, where $\mathbb{P}(\hat{b} = b) = \langle b|u\rho u^\dagger|b\rangle$,

6. Let $M_{\mathfrak{U}}(\rho) := \mathbb{E}(u^\dagger|\hat{b}\rangle\langle\hat{b}|u) = \mathbb{E}_{u \sim \mathfrak{U}} \sum_{b \in \{0,1\}^n} \langle b|u\rho u^\dagger|b\rangle u^\dagger|b\rangle\langle b|u$,

7. Construct a classical snapshot by computing $\hat{\rho} = M_{\mathfrak{U}}^{-1}(u^\dagger|\hat{b}\rangle\langle\hat{b}|u)$.

State $\hat{\rho}$ is called classical shadow, which has a unit trace, but need not be positive semi-definite. We note that tomographically completeness ensures that $M_{\mathfrak{U}}^{-1}$ exists.

Classical shadow is a useful tool in numerical experiments based on random Clifford measurements [40]. We know that a stabilizer circuit can be simulated efficiently in $O(n^2)$ time by applying the algorithm based on tableau representation. This algorithm enables us to simulate systems efficiently with even more than 160 qubits in practice. To test the performance of predicting with classical shadows, we need to implement repeatedly the following efficient protocol.

1. Sample a Clifford unitary $u$ from $C_n$ by applying the algorithm proposed in [41].

2. The action of $u$ on X- and Z-type operators is fully characterized by parameters $(\alpha, \beta, \gamma, \delta, r, t)$ as follows

$$uX_j u^\dagger = (-1)^{r_j} \prod_{i=1}^{n} X_i^{\alpha_{ji}} Z_i^{\beta_{ji}}, \quad uZ_j u^\dagger = (-1)^{t_j} \prod_{i=1}^{n} X_i^{\gamma_{ji}} Z_i^{\delta_{ji}},$$

where $j = 1, 2, ..., n$ and $r_j, t_j, \alpha_{ji}, \beta_{ji}, \gamma_{ji}$ and $\delta_{ji} \in \{0, 1\}$.

3. A parameterized unitary $u$ by $(\alpha, \beta, \gamma, \delta, r, t)$ can be applied on stabilizer states by changing the stabilizer generators and the destabilizer generators.

4. A computational basis measurement can be simulated by the improved algorithm based on tableau representation.
# 9 Discussion

In this section, we present some research lines as follows.

* The family of quantum low-density parity-check (qLDPC) codes is a subclass of CSS codes, which plays a central role in classical error correction [43]. Existence of qLDPC codes whose minimal distance scales linearly with the number of qubits is one of the major open problems in quantum information [44]. Furthermore, to see a set of open problems on geometric and combinatorial aspects of qLDPC codes, refer to [45].

* The *XP* stabilizer formalism is an extension of the stabilizer formalism which includes fractional $\frac{2\pi}{n}$ rotations around the *Z* axis, where *n* is an integer. There is an equivalence between *XP* stabilizer states and weighted hypergraph states. To see a set of open problems in the *XP* stabilizer formalism, refer to [46].

# 10 Conclusion

The process of measuring a Pauli observable is discussed. It is shown that there exists a stabilizer witness consisting of a full set of generators, which is coarser than the GHZ witness. Moreover, with less than *n* generators, it is not possible to detect genuine *n*-qubit entanglement. A stabilizer code is constructed from a given linear code. The evolution of a quantum system in the presence of noise is discussed by giving the error-correcting criteria. Moreover, the process of syndrome extraction for correctable errors is discussed. By applying logical bases, it is shown that any stabilizer code is unitarily equivalent to a trivial code. Given a graph code, its stabilizer generators are obtained. By applying the cleaning lemma, the distance of a stabilizer code embeddable in a lattice is given. By applying stabilizer matrices, the runtime of simulating stabilizer gates is obtained. The Knill-Gottesman theorem on classical simulation of stabilizer circuits is discussed. It is shown that given a full tableau, every Pauli product can be decomposed efficiently into a product of a stabilizer and a destabilizer. An algorithm for updating global phases by applying frame representation is given. Resolution of a quantum channel into a sum of stabilizer channels is shown. A family of capacity achieving stabilizer codes is applied to obtain the capacity of the quantum erasure channel. An algorithm for constructing classical shadow is given.

**Acknowledgement**. This work has been performed in the Laboratory of Combinatorial and Geometric Structures, School of Applied Mathematics and Informatics, MIPT. I would like to thank Prof. Andrei M. Raigorodskii, Dr. Andrey B. Kupavskii and my colleagues.
# 9 Discussion

In this section, we present some research lines as follows.

* The family of quantum low-density parity-check (qLDPC) codes is a subclass of CSS codes, which plays a central role in classical error correction [43]. Existence of qLDPC codes whose minimal distance scales linearly with the number of qubits is one of the major open problems in quantum information [44]. Furthermore, to see a set of open problems on geometric and combinatorial aspects of qLDPC codes, refer to [45].

* The *XP* stabilizer formalism is an extension of the stabilizer formalism which includes fractional $\frac{2\pi}{n}$ rotations around the *Z* axis, where *n* is an integer. There is an equivalence between *XP* stabilizer states and weighted hypergraph states. To see a set of open problems in the *XP* stabilizer formalism, refer to [46].

# 10 Conclusion

The process of measuring a Pauli observable is discussed. It is shown that there exists a stabilizer witness consisting of a full set of generators, which is coarser than the GHZ witness. Moreover, with less than *n* generators, it is not possible to detect genuine *n*-qubit entanglement. A stabilizer code is constructed from a given linear code. The evolution of a quantum system in the presence of noise is discussed by giving the error-correcting criteria. Moreover, the process of syndrome extraction for correctable errors is discussed. By applying logical bases, it is shown that any stabilizer code is unitarily equivalent to a trivial code. Given a graph code, its stabilizer generators are obtained. By applying the cleaning lemma, the distance of a stabilizer code embeddable in a lattice is given. By applying stabilizer matrices, the runtime of simulating stabilizer gates is obtained. The Knill-Gottesman theorem on classical simulation of stabilizer circuits is discussed. It is shown that given a full tableau, every Pauli product can be decomposed efficiently into a product of a stabilizer and a destabilizer. An algorithm for updating global phases by applying frame representation is given. Resolution of a quantum channel into a sum of stabilizer channels is shown. A family of capacity achieving stabilizer codes is applied to obtain the capacity of the quantum erasure channel. An algorithm for constructing classical shadow is given.

**Acknowledgement.** This work has been performed in the Laboratory of Combinatorial and Geometric Structures, School of Applied Mathematics and Informatics, MIPT. I would like to thank Prof. Andrei M. Raigorodskii, Dr. Andrey B. Kupavskii and my colleagues.
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