System #0

You are a physics research assistant specializing in solving complex, research-level problems using precise, step-by-step reasoning.

Input Problems will be provided in Markdown format.

Output (Markdown format)

Step-by-Step Derivation - Show every non-trivial step in the solution. Justify steps using relevant physical laws, theorems, or mathematical identities.
Mathematical Typesetting - Use LaTeX for all mathematics: $...$ for inline expressions, $$...$$ for display equations.
Conventions and Units - Follow the unit system and conventions specified in the problem.
Final Answer - At the end of the solution, start a new line with “Final Answer:”, and present the final result.

For final answers involving values, follow the precision requirements specified in the problem. If no precision is specified: - If an exact value is possible, provide it (e.g., \$\sqrt(2)\$, \$\pi/4\$). - If exact form is not feasible, retain at least 12 significant digits in the result.
Formatting Compliance - If the user requests a specific output format (e.g., code, table), provide the final answer accordingly.

User #1

Problem setup:

In quantum error correction, you encode quantum states into logical states made of many qubits in order to improve their resilience to errors. In quantum error detection, you do the same but can only detect the presence of errors and not correct them. In this problem, we will consider a single [[4,2,2]] quantum error detection code, which encodes two logical qubits into four physical qubits, and investigate how robust logical quantum operations in this code are to quantum errors.

Our convention is that the four physical qubits in the [[4,2,2]] code are labelled 0,1,2,3. The two logical qubits are labelled A and B. The stabilizers are $XXXX$ and $ZZZZ$, where $X$ and $Z$ are Pauli matrices. The logical $X$ and $Z$ operators on the two qubits are $X_A = XIXI$, $X_B=XXII$, $Z_A = ZZII$, $Z_B = ZIZI$, up to multiplication by stabilizers.

We will consider different state preparation circuits consisting of controlled not $CNOT_{ij}$ gates, where $CNOT_{ij}$ has control qubit $i$ and target qubit $j$. As a simple model of quantum errors in hardware, we will suppose that each $CNOT_{ij}$ gate in the circuit has a two qubit depolarizing error channel following it that produces one of the 15 non-identity two-qubit Paulis with equal probability $p/15$. The probability $p$ indicates the probability of an error in a single two-qubit gate. We will assess the logical infidelity of certain state preparation protocols as a function of the physical infidelity $p$.

Checkpoint 1

Suppose that we wish to prepare a logical two-qubit GHZ state $(|00\rangle_{AB}+|11\rangle_{AB})/\sqrt{2}$ in the [[4,2,2]] code. To do so, we use the following state preparation circuit:

\[ (CNOT_{03}) (H_0) (CNOT_{21}) (H_2). \]

Note that this equation is written in matrix multiplication order, while the quantum operations in the circuit occur in the reverse order (from right-to-left in the above equation). $H$ is a single-qubit Hadamard gate.

What is the physical state fidelity of the final physical 4-qubit state at the end of the circuit as a function of the two-qubit gate error rate $p$?

Assistant #2

Reasoning (model)

Fidelity of a Logical GHZ State with Depolarizing Errors

My task is to determine the fidelity of a logical GHZ state prepared using a specific quantum circuit, taking into account the impact of depolarizing errors that occur after each CNOT gate. The circuit is $(CNOT_{03}) (H_0) (CNOT_{21}) (H_2)$. My field expertise means I’m comfortable navigating the intricacies of quantum error correction.

First, I had to understand the target state and its physical representation. The goal is to prepare a logical GHZ state $(|00\rangle_{AB}+|11\rangle_{AB})/\sqrt{2}$ within the [[4,2,2]] code. I know that codewords within this code must be $+1$ eigenstates of the stabilizer operators. Using the stabilizer formalism, I verified that the logical state $|00\rangle_{AB}$ is correctly represented by the state $|\psi_0\rangle = \frac{1}{\sqrt{2}} (|0000\rangle + |1111\rangle)$. I then had to use the logical operators $X_A = X_0I_1X_2I_3$, $X_B=X_0X_1I_2I_3$, $Z_A = Z_0Z_1I_2I_3$, $Z_B = Z_0I_1Z_2I_3$ to identify $|11\rangle_L$.

Next, I determined the ideal circuit, obtaining the correct physical state $|\psi_{target}\rangle = \frac{1}{2} (|0000\rangle + |1111\rangle + |0110\rangle + |1001\rangle)$ through gate by gate simulation.

Now for the error model. Each CNOT gate is followed by a two-qubit depolarizing channel $\mathcal{D}(\rho) = (1-p) \rho + \frac{p}{15} \sum_{P \neq I} P \rho P^\dagger$. This produces one of the 15 non-identity two-qubit Paulis with probability $p/15$. My goal is to find the fidelity $F = \langle \psi_{target} | \rho_{final} | \psi_{target} \rangle$.

Calculating the fidelity requires some key simplifications. Since $H_0$ and $CNOT_{03}$ commute with the support of $P_{21}$ and $P_{03}$ respectively, the fidelity can be expressed in terms of the number of stabilizers, and how the Pauli group elements from the error channels transform. Using the property that if $P$ is a Pauli operator, $\langle\psi|P|\psi\rangle = 0$ unless $P$ is in the stabilizer group of the state, I can compute the fidelity as $F = 1 - \frac{8}{5}p + \frac{143}{225}p^2$. The key result here is that fidelity decays with $p^2$.

My work is now complete.

The problem asks for the physical state fidelity of the final 4-qubit state, which is a logical GHZ state, as a function of the two-qubit gate error rate $p$. Errors occur after each CNOT gate, modeled by a two-qubit depolarizing channel.

First, let’s identify the target logical GHZ state in the physical qubit basis. The logical GHZ state is $(|00\rangle_{AB}+|11\rangle_{AB})/\sqrt{2}$. For the [[4,2,2]] code, the logical basis states in the physical qubit basis are: $|00\rangle_L = \frac{1}{\sqrt{2}}(|0000\rangle + |1111\rangle)$ $|01\rangle_L = \frac{1}{\sqrt{2}}(|0011\rangle + |1100\rangle)$ $|10\rangle_L = \frac{1}{\sqrt{2}}(|0101\rangle + |1010\rangle)$ $|11\rangle_L = \frac{1}{\sqrt{2}}(|0110\rangle + |1001\rangle)$ Thus, the target physical state $|\psi_{target}\rangle$ is:

\[ |\psi_{target}\rangle = \frac{1}{\sqrt{2}} (|00\rangle_L + |11\rangle_L) = \frac{1}{2} (|0000\rangle + |1111\rangle + |0110\rangle + |1001\rangle) \]

Next, let’s verify that the given circuit produces this state in the absence of errors. We start from the initial state $|\psi_{init}\rangle = |0000\rangle$. The circuit is $(CNOT_{03}) (H_0) (CNOT_{21}) (H_2)$. Operations are applied right-to-left.

Apply $H_2$: $|\psi_1\rangle = H_2 |0000\rangle = \frac{1}{\sqrt{2}}(|0000\rangle + |0010\rangle)$.
Apply $CNOT_{21}$ (control 2, target 1): $|\psi_2\rangle = CNOT_{21} |\psi_1\rangle = \frac{1}{\sqrt{2}}(CNOT_{21}|0000\rangle + CNOT_{21}|0010\rangle) = \frac{1}{\sqrt{2}}(|0000\rangle + |0110\rangle)$.
Apply $H_0$: $|\psi_3\rangle = H_0 |\psi_2\rangle = \frac{1}{2}(H_0|0000\rangle + H_0|0110\rangle) = \frac{1}{2}((|0000\rangle + |1000\rangle) + (|0110\rangle + |1110\rangle))$. $|\psi_3\rangle = \frac{1}{2}(|0000\rangle + |1000\rangle + |0110\rangle + |1110\rangle)$.
Apply $CNOT_{03}$ (control 0, target 3): $|\psi_{final}\rangle = CNOT_{03} |\psi_3\rangle = \frac{1}{2}(CNOT_{03}|0000\rangle + CNOT_{03}|1000\rangle + CNOT_{03}|0110\rangle + CNOT_{03}|1110\rangle)$. $|\psi_{final}\rangle = \frac{1}{2}(|0000\rangle + |1001\rangle + |0110\rangle + |1111\rangle)$. This matches the target state $|\psi_{target}\rangle$.

Now, we incorporate errors. A two-qubit depolarizing error channel $\mathcal{D}_{ij}(\rho)$ follows each $CNOT_{ij}$ gate. The channel is defined as:

\[ \mathcal{D}_{ij}(\rho) = (1-p)\rho + \sum_{k=1}^{15} \frac{p}{15} P_k \rho P_k^\dagger \]

where $P_k$ are the 15 non-identity two-qubit Pauli operators acting on qubits $i$ and $j$.

Let $U_A = H_2$, $U_B = CNOT_{21}$, $U_C = H_0$, $U_D = CNOT_{03}$. The state evolution is: $\rho_0 = |0000\rangle\langle0000|$ $\rho_1 = U_A \rho_0 U_A^\dagger$ $\rho_2 = U_B \rho_1 U_B^\dagger$ $\rho_3 = \mathcal{D}_{21}(\rho_2)$ (error after $CNOT_{21}$ on qubits 2,1) $\rho_4 = U_C \rho_3 U_C^\dagger$ $\rho_5 = U_D \rho_4 U_D^\dagger$ $\rho_{final} = \mathcal{D}_{03}(\rho_5)$ (error after $CNOT_{03}$ on qubits 0,3)

We want to compute the fidelity $F = \langle \psi_{target} | \rho_{final} | \psi_{target} \rangle$. Let $|\psi\rangle = |\psi_{target}\rangle$. Let $\rho_{ideal}$ be the state if no errors occur, so $\rho_{ideal} = |\psi\rangle\langle\psi|$. The key observation for this problem is how the error operators propagate through the ideal gates.

Let’s look at the first error channel $\mathcal{D}_{21}$. It acts on qubits 1 and 2. The subsequent operations are $H_0$ (on qubit 0) and $CNOT_{03}$ (on qubits 0 and 3). These operations act on qubits disjoint from qubits 1 and 2. Therefore, $H_0$ commutes with any $P_{21}$ (Pauli on qubits 1,2): $H_0 P_{21} H_0^\dagger = P_{21}$. Similarly, $CNOT_{03}$ commutes with any $P_{21}$: $CNOT_{03} P_{21} CNOT_{03}^\dagger = P_{21}$.

So, the density matrix $\rho_5$ becomes:

\[ \rho_5 = U_D U_C \mathcal{D}_{21}(\rho_2) U_C^\dagger U_D^\dagger = (1-p) U_D U_C \rho_2 U_C^\dagger U_D^\dagger + \frac{p}{15} \sum_{P_{21}} U_D U_C P_{21} \rho_2 P_{21}^\dagger U_C^\dagger U_D^\dagger \]

Since $U_D U_C P_{21} U_C^\dagger U_D^\dagger = P_{21}$:

\[ \rho_5 = (1-p) |\psi\rangle\langle\psi| + \frac{p}{15} \sum_{P_{21}} P_{21} |\psi\rangle\langle\psi| P_{21}^\dagger \]

Now apply the second error channel $\mathcal{D}_{03}$ (on qubits 0,3):

\[ \rho_{final} = \mathcal{D}_{03}(\rho_5) = (1-p)\rho_5 + \frac{p}{15} \sum_{P_{03}} P_{03} \rho_5 P_{03}^\dagger \]

Substituting $\rho_5$:

\[ \rho_{final} = (1-p) \left[ (1-p) |\psi\rangle\langle\psi| + \frac{p}{15} \sum_{P_{21}} P_{21} |\psi\rangle\langle\psi| P_{21}^\dagger \right] + \frac{p}{15} \sum_{P_{03}} P_{03} \left[ (1-p) |\psi\rangle\langle\psi| + \frac{p}{15} \sum_{P_{21}} P_{21} |\psi\rangle\langle\psi| P_{21}^\dagger \right] P_{03}^\dagger \]

The fidelity $F = \langle \psi | \rho_{final} | \psi \rangle$ is:

\[ F = (1-p)^2 \langle \psi | \psi\rangle\langle\psi| | \psi \rangle + (1-p)\frac{p}{15} \sum_{P_{21}} \langle \psi | P_{21} |\psi\rangle\langle\psi| P_{21}^\dagger | \psi \rangle + \frac{p}{15}(1-p) \sum_{P_{03}} \langle \psi | P_{03} |\psi\rangle\langle\psi| P_{03}^\dagger | \psi \rangle + \frac{p^2}{15^2} \sum_{P_{03}, P_{21}} \langle \psi | P_{03} P_{21} |\psi\rangle\langle\psi| P_{21}^\dagger P_{03}^\dagger | \psi \rangle \]

This simplifies to:

\[ F = (1-p)^2 + (1-p)\frac{p}{15} \sum_{P_{21}} |\langle \psi | P_{21} |\psi\rangle|^2 + \frac{p}{15}(1-p) \sum_{P_{03}} |\langle \psi | P_{03} |\psi\rangle|^2 + \frac{p^2}{15^2} \sum_{P_{03}, P_{21}} |\langle \psi | P_{03} P_{21} |\psi\rangle|^2 \]

The target state $|\psi\rangle$ is a stabilizer state. For a stabilizer state, $\langle\psi|P|\psi\rangle = \pm 1$ if $P$ is in the stabilizer group of $|\psi\rangle$, and $0$ otherwise. Thus, $|\langle\psi|P|\psi\rangle|^2 = 1$ if $P \in \mathrm{Stab}(|\psi\rangle)$, and $0$ if $P \notin \mathrm{Stab}(|\psi\rangle)$.

The generators of the stabilizer group for $|\psi\rangle = \frac{1}{2} (|0000\rangle + |1111\rangle + |0110\rangle + |1001\rangle)$ are: $g_1 = X_0X_1X_2X_3$ (code stabilizer $S_1$) $g_2 = Z_0Z_1Z_2Z_3$ (code stabilizer $S_2$) $g_3 = X_A X_B = (X_0I_1X_2I_3)(X_0X_1I_2I_3) = I_0X_1X_2I_3$ (logical operator for GHZ state) $g_4 = Z_A Z_B = (Z_0Z_1I_2I_3)(Z_0I_1Z_2I_3) = I_0Z_1Z_2I_3$ (logical operator for GHZ state) These 4 generators are mutually commuting and independent, generating a stabilizer group $\mathcal{S}_\psi$ of $2^4=16$ elements.

Let’s evaluate the sums:

$\sum_{P_{21}} |\langle \psi | P_{21} |\psi\rangle|^2$: $P_{21}$ are the 15 non-identity Pauli operators of the form $I_0 P_1 P_2 I_3$. We need to count how many of these are in $\mathcal{S}_\psi$.
- $g_3 = I_0X_1X_2I_3$. This is a $P_{21}$ type operator.
- $g_4 = I_0Z_1Z_2I_3$. This is a $P_{21}$ type operator.
- $g_3g_4 = (I_0X_1X_2I_3)(I_0Z_1Z_2I_3) = I_0Y_1Y_2I_3$. This is a $P_{21}$ type operator. There are no other elements in $\mathcal{S}_\psi$ that are of the form $I_0P_1P_2I_3$. So, the sum is $3$.
$\sum_{P_{03}} |\langle \psi | P_{03} |\psi\rangle|^2$: $P_{03}$ are the 15 non-identity Pauli operators of the form $P_0 I_1 I_2 P_3$. We need to count how many of these are in $\mathcal{S}_\psi$.
- $g_1g_3 = (X_0X_1X_2X_3)(I_0X_1X_2I_3) = X_0I_1I_2X_3$. This is a $P_{03}$ type operator.
- $g_2g_4 = (Z_0Z_1Z_2Z_3)(I_0Z_1Z_2I_3) = Z_0I_1I_2Z_3$. This is a $P_{03}$ type operator.
- $(g_1g_3)(g_2g_4) = (X_0I_1I_2X_3)(Z_0I_1I_2Z_3) = Y_0I_1I_2Y_3$. This is a $P_{03}$ type operator. There are no other elements in $\mathcal{S}_\psi$ that are of the form $P_0I_1I_2P_3$. So, the sum is $3$.
$\sum_{P_{03}, P_{21}} |\langle \psi | P_{03} P_{21} |\psi\rangle|^2$: This sum is over $15 \times 15 = 225$ terms. A term $P_{03}P_{21}$ is of the form $P_0P_1P_2P_3$. We need to count how many of these products are in $\mathcal{S}_\psi$, subject to the conditions that $P_{03} \ne I_4$ and $P_{21} \ne I_4$. This means $P$ cannot be $I_4$, and $P$ cannot be an element of $\mathcal{S}_\psi$ that acts only on qubits 0,3 (like $X_0I_1I_2X_3$) or only on qubits 1,2 (like $I_0X_1X_2I_3$). Let’s list the 16 elements of $\mathcal{S}_\psi$:
1. $I_4$: Excluded because $P_{03} \ne I_4$ and $P_{21} \ne I_4$.
2. $g_1 = X_0X_1X_2X_3$: $(X_0X_3)(X_1X_2)$. Both parts are non-identity. Count = 1.
3. $g_2 = Z_0Z_1Z_2Z_3$: $(Z_0Z_3)(Z_1Z_2)$. Both parts are non-identity. Count = 1.
4. $g_3 = I_0X_1X_2I_3$: $P_{03}=I_4$. Excluded.
5. $g_4 = I_0Z_1Z_2I_3$: $P_{03}=I_4$. Excluded.
6. $g_1g_2 = Y_0Y_1Y_2Y_3$: $(Y_0Y_3)(Y_1Y_2)$. Both parts are non-identity. Count = 1.
7. $g_1g_3 = X_0I_1I_2X_3$: $P_{21}=I_4$. Excluded.
8. $g_1g_4 = X_0Y_1Y_2X_3$: $(X_0X_3)(Y_1Y_2)$. Both parts are non-identity. Count = 1.
9. $g_2g_3 = Z_0X_1X_2Z_3$: $(Z_0Z_3)(X_1X_2)$. Both parts are non-identity. Count = 1.
10. $g_2g_4 = Z_0I_1I_2Z_3$: $P_{21}=I_4$. Excluded.
11. $g_3g_4 = I_0Y_1Y_2I_3$: $P_{03}=I_4$. Excluded.
12. $g_1g_2g_3 = Y_0I_1I_2Y_3$: $P_{21}=I_4$. Excluded.
13. $g_1g_2g_4 = Y_0X_1X_2Y_3$: $(Y_0Y_3)(X_1X_2)$. Both parts are non-identity. Count = 1.
14. $g_1g_3g_4 = X_0Z_1Z_2X_3$: $(X_0X_3)(Z_1Z_2)$. Both parts are non-identity. Count = 1.
15. $g_2g_3g_4 = Z_0Y_1Y_2Z_3$: $(Z_0Z_3)(Y_1Y_2)$. Both parts are non-identity. Count = 1. There are 8 such operators. So, the sum is $8$.

Substitute these values back into the fidelity expression:

\[ F = (1-p)^2 + (1-p)\frac{p}{15}(3) + \frac{p}{15}(1-p)(3) + \frac{p^2}{15^2}(8) \]

\[ F = (1-2p+p^2) + \frac{3p(1-p)}{15} + \frac{3p(1-p)}{15} + \frac{8p^2}{225} \]

\[ F = (1-2p+p^2) + \frac{2p(1-p)}{5} + \frac{8p^2}{225} \]

\[ F = 1-2p+p^2 + \frac{2p}{5} - \frac{2p^2}{5} + \frac{8p^2}{225} \]

Combine terms:

\[ F = 1 + \left(-2 + \frac{2}{5}\right)p + \left(1 - \frac{2}{5} + \frac{8}{225}\right)p^2 \]

\[ F = 1 + \left(-\frac{10}{5} + \frac{2}{5}\right)p + \left(\frac{225}{225} - \frac{90}{225} + \frac{8}{225}\right)p^2 \]

\[ F = 1 - \frac{8}{5}p + \frac{143}{225}p^2 \]

The final answer is $\boxed{1 - \frac{8}{5}p + \frac{143}{225}p^2}$.

User #3

Checkpoint 2

Suppose that we wish to prepare a logical two-qubit GHZ state $(|00\rangle_{AB}+|11\rangle_{AB})/\sqrt{2}$ in the [[4,2,2]] code. To do so, we use the following state preparation circuit:

\[ (CNOT_{03}) (H_0) (CNOT_{21}) (H_2). \]

What is the logical state fidelity of the final 2-qubit logical state at the end of the circuit as a function of the two-qubit gate error rate $p$, assuming the state is post-selected on all detectable errors in the code?

Assistant #4

Reasoning (model)

Summary of My Reasoning:

I’m being asked to calculate the logical state fidelity of a prepared logical GHZ state after applying a post-selection process that discards states with detectable errors. My first task is to project the physical state onto the code space, which is a key step in analyzing the logical fidelity.

I’ll start with the ideal target logical GHZ state and its corresponding physical representation, as derived in a previous subproblem. The density matrix, $\rho_{final}$, incorporates errors from a two-qubit depolarizing channel acting on specific qubits. I need to project this noisy state onto the code space defined by the [[4,2,2]] code’s stabilizers, $S_1$ and $S_2$. The code space is the +1 eigenspace of both stabilizers, and to do the projection I’ll use the operator $P_{\mathcal{C}} = \frac{1}{4} (I+S_1)(I+S_2)$.

The logical fidelity, $F_L$, is then the overlap between the ideal state and the post-selected state. Since the logical GHZ state is already in the code space, the numerator of the fidelity calculation simplifies to the physical fidelity from the prior subproblem, which is also the numerator I’m aiming for. My work is then to calculate the trace of the post-selected density matrix $\mathrm{Tr}(\rho')$ where $\rho' = P_{\mathcal{C}} \rho_{final} P_{\mathcal{C}}$.

To simplify things, I’ll break $\mathrm{Tr}(\rho')$ down, and analyze each term in the density matrix. I need to calculate $\mathrm{Tr}(\rho_{final} P_{\mathcal{C}})$. I realized I need to check whether $E|\psi\rangle$ is a vector in the code space if $E$ maps a state outside the code space. If $P_{21}|\psi\rangle$ is orthogonal to the code space, then its contribution is zero. If the error is not detectable, this will change.

A key part of the problem lies in analyzing how different Pauli error operators behave with respect to the stabilizers. The given errors are two-qubit errors which is where the analysis focuses. Crucially, I found that all the considered two-qubit errors ($P_{21}$ and $P_{03}$) actually commute with the code’s stabilizers, even though they can change the qubits, and therefore map the initial state back into the code space. I had to go over the computations multiple times to convince myself of this, as it sounds like an unrealistic simplification of real error models. I carefully checked the syndrome condition to ensure that the errors would not result in the post-selection mechanism being used, which would give a more practical post-selected fidelity of 1.

The fact that the errors do not generate a syndrome (because they map the state back into the code space) is crucial. This means that the post-selection step doesn’t actually discard any states in this specific error model and state preparation. This simplified my calculation significantly. It means the trace becomes equal to $1$. In the context of “post-selection on all detectable errors”, there are no detectable errors in this case.

With this crucial realization, I conclude that the final, post-selected state is identical to the final noisy state, $\rho'=\rho_{final}$. The trace of this normalized state is equal to 1. It is worth keeping in mind that these error terms were very specific, so I need to maintain clarity on the underlying reasoning.

The logical fidelity is simply, the physical fidelity. The formula for logical fidelity is $\frac{\langle\psi_L|\rho_L|\psi_L\rangle}{\mathrm{Tr}(\rho_L)}$ where $\rho_L$ is the post-selected logical state. Since my computation is very specific, this comes to one. I needed to verify all terms were computed correctly.

The overall result that the fidelity is 1 seems counterintuitive at first – that the fidelity is perfect after post-selection and for this type of errors. But it’s a direct consequence of the problem setup, where the error operators commute with the stabilizers, which indicates a noiseless subsystem (as I verified). All errors are undetected in this scenario.

To determine the logical state fidelity of the final 2-qubit logical state, post-selected on all detectable errors, we need to calculate two quantities:

The fidelity of the final physical state $\rho_{final}$ with the target ideal state $|\psi_{target}\rangle$, denoted as $F = \langle\psi_{target}|\rho_{final}|\psi_{target}\rangle$.
The trace of the projected state onto the code space, $\mathrm{Tr}(P_{\mathcal{C}}\rho_{final}P_{\mathcal{C}})$, which represents the probability of successful post-selection.

The logical state fidelity $F_L$ is then given by $F_L = \frac{F}{\mathrm{Tr}(P_{\mathcal{C}}\rho_{final}P_{\mathcal{C}})}$.

Let’s first clarify the ideal target state and its stabilizer group. The target logical GHZ state is $|\psi_L\rangle = \frac{1}{\sqrt{2}}(|00\rangle_{AB}+|11\rangle_{AB})$. Its physical representation is:

\[ |\psi_{target}\rangle = \frac{1}{2} (|0000\rangle + |1111\rangle + |0110\rangle + |1001\rangle) \]

The stabilizer group $\mathcal{S}_\psi$ for this state is generated by: $g_1 = X_0X_1X_2X_3$ (code stabilizer $S_1$) $g_2 = Z_0Z_1Z_2Z_3$ (code stabilizer $S_2$) $g_3 = I_0X_1X_2I_3$ (logical operator $X_A X_B = (X_0I_1X_2I_3)(X_0X_1I_2I_3)$ for $|00\rangle_L+|11\rangle_L$) $g_4 = I_0Z_1Z_2I_3$ (logical operator $Z_A Z_B = (Z_0Z_1I_2I_3)(Z_0I_1Z_2I_3)$ for $|00\rangle_L+|11\rangle_L$)

The final physical density matrix $\rho_{final}$ was derived in Subproblem 1. It is given by:

\[ \rho_{final} = (1-p)^2 |\psi\rangle\langle\psi| + (1-p)\frac{p}{15} \sum_{P_{21} \neq I} P_{21} |\psi\rangle\langle\psi| P_{21}^\dagger + \frac{p}{15}(1-p) \sum_{P_{03} \neq I} P_{03} |\psi\rangle\langle\psi| P_{03}^\dagger + \frac{p^2}{15^2} \sum_{P_{03} \neq I, P_{21} \neq I} P_{03} P_{21} |\psi\rangle\langle\psi| P_{21}^\dagger P_{03}^\dagger \]

where $P_{21}$ are the 15 non-identity Pauli operators on qubits 1 and 2, and $P_{03}$ are the 15 non-identity Pauli operators on qubits 0 and 3.

The fidelity $F = \langle\psi|\rho_{final}|\psi\rangle$ requires computing $|\langle\psi|E|\psi\rangle|^2$ for each error $E$. This value is 1 if $E \in \mathcal{S}_\psi$ and 0 otherwise.

Let’s re-evaluate the counts for the terms in $F$, as there was a slight error in the initial calculation of Subproblem 1.

Number of $P_{21} \neq I$ (Pauli operators on qubits 1,2 only) that are in $\mathcal{S}_\psi$: These are $g_3 = I_0X_1X_2I_3$, $g_4 = I_0Z_1Z_2I_3$, and $g_3g_4 = I_0Y_1Y_2I_3$. There are 3 such operators.
Number of $P_{03} \neq I$ (Pauli operators on qubits 0,3 only) that are in $\mathcal{S}_\psi$: These are $g_1g_3 = X_0I_1I_2X_3$, $g_2g_4 = Z_0I_1I_2Z_3$, and $(g_1g_3)(g_2g_4) = Y_0I_1I_2Y_3$. There are 3 such operators. (This corresponds to $(g_1g_2)g_3$ in my derivation path, which is $Y_0Y_1Y_2Y_3 g_3 = Y_0X_1Y_2Y_3$ which is not $Y_0I_1I_2Y_3$. Let’s be careful. The elements of $\mathcal{S}_\psi$ that are $P_{03}$ type are $X_0X_3$, $Z_0Z_3$, $Y_0Y_3$ acting on the logical state in the proper way. More specifically, these are $S_1(I_1I_2)S_1 = (X_0X_3)X_1X_2$. The relevant operators that are in $\mathcal{S}_\psi$ and are of the form $P_0I_1I_2P_3$ are $X_0I_1I_2X_3$, $Z_0I_1I_2Z_3$, and $Y_0I_1I_2Y_3$. These are $g_1g_3$, $g_2g_4$, and $g_1g_2g_3$ as listed in the thought process). So this count is 3.
Number of $P_{03}P_{21}$ (Pauli operators on all 4 qubits, with non-identity parts on (0,3) and (1,2)) that are in $\mathcal{S}_\psi$: These are products $P_{03}'P_{21}'$ where $P_{03}' \in \{X_0X_3, Y_0Y_3, Z_0Z_3\}$ and $P_{21}' \in \{X_1X_2, Y_1Y_2, Z_1Z_2\}$. There are $3 \times 3 = 9$ such operators.

Substituting these counts into the fidelity expression:

\[ F = (1-p)^2 \times 1 + (1-p)\frac{p}{15}(3) + \frac{p}{15}(1-p)(3) + \frac{p^2}{15^2}(9) \]

\[ F = (1-2p+p^2) + \frac{6p(1-p)}{15} + \frac{9p^2}{225} \]

\[ F = (1-2p+p^2) + \frac{2p(1-p)}{5} + \frac{p^2}{25} \]

\[ F = 1-2p+p^2 + \frac{2}{5}p - \frac{2}{5}p^2 + \frac{1}{25}p^2 \]

\[ F = 1 + \left(-2 + \frac{2}{5}\right)p + \left(1 - \frac{2}{5} + \frac{1}{25}\right)p^2 \]

\[ F = 1 - \frac{8}{5}p + \left(\frac{25 - 10 + 1}{25}\right)p^2 \]

\[ F = 1 - \frac{8}{5}p + \frac{16}{25}p^2 \]

Next, we calculate the trace of the post-selected state. The projector onto the code space is $P_{\mathcal{C}} = \frac{1}{4}(I+S_1)(I+S_2)$. We need to calculate $\mathrm{Tr}(P_{\mathcal{C}}\rho_{final}P_{\mathcal{C}})$. For any error $E$, the term $\mathrm{Tr}(P_{\mathcal{C}} E |\psi\rangle\langle\psi| E^\dagger P_{\mathcal{C}})$ evaluates to 1 if $E|\psi\rangle$ is in the code space, and 0 otherwise. $E|\psi\rangle$ is in the code space if and only if $E$ commutes with both code stabilizers $S_1=X_0X_1X_2X_3$ and $S_2=Z_0Z_1Z_2Z_3$.

Let’s re-evaluate which error operators $E$ map the state $|\psi\rangle$ into the code space:

For $P_{21}$ (operators of the form $I_0P_1P_2I_3$): $P_1P_2$ must commute with $X_1X_2$ (for $S_1$) and $Z_1Z_2$ (for $S_2$). These are the 4 operators $\{I_1I_2, X_1X_2, Y_1Y_2, Z_1Z_2\}$. Since $P_{21} \neq I$, there are 3 such operators ($I_0X_1X_2I_3, I_0Y_1Y_2I_3, I_0Z_1Z_2I_3$). These are precisely the elements of $\mathcal{S}_\psi$ on qubits 1,2.
For $P_{03}$ (operators of the form $P_0I_1I_2P_3$): $P_0P_3$ must commute with $X_0X_3$ (for $S_1$) and $Z_0Z_3$ (for $S_2$). These are the 4 operators $\{I_0I_3, X_0X_3, Y_0Y_3, Z_0Z_3\}$. Since $P_{03} \neq I$, there are 3 such operators ($X_0I_1I_2X_3, Y_0I_1I_2Y_3, Z_0I_1I_2Z_3$). These are precisely the elements of $\mathcal{S}_\psi$ on qubits 0,3.
For $P_{03}P_{21}$ (operators of the form $P_0P_1P_2P_3$): $(P_0P_3)$ must commute with $X_0X_3$ and $Z_0Z_3$, and $(P_1P_2)$ must commute with $X_1X_2$ and $Z_1Z_2$. This gives $4 \times 4 = 16$ possible combinations of $(P_0P_3, P_1P_2)$. We exclude the case where $P_0P_3=I_0I_3$ and $P_1P_2=I_1I_2$ (the identity error). Also, we only count terms where $P_{03} \ne I$ and $P_{21} \ne I$. The number of such errors that map to the code space is $(4-1)(4-1)=3 \times 3 = 9$.

All operators $E$ that map $|\psi\rangle$ into the code space are in fact stabilizers of $|\psi\rangle$ (i.e., $E \in \mathcal{S}_\psi$). This means that for these specific error operators, $E|\psi\rangle = \pm |\psi\rangle$. Therefore, for any error $E$ that maps to the code space, $\mathrm{Tr}(P_{\mathcal{C}} E |\psi\rangle\langle\psi| E^\dagger P_{\mathcal{C}}) = \langle\psi| E^\dagger P_{\mathcal{C}} E |\psi\rangle = \langle\psi| E^\dagger E |\psi\rangle = \langle\psi|\psi\rangle = 1$. For any error $E$ that maps outside the code space, $\mathrm{Tr}(P_{\mathcal{C}} E |\psi\rangle\langle\psi| E^\dagger P_{\mathcal{C}}) = 0$.

So, the trace of the projected state is:

\[ \mathrm{Tr}(P_{\mathcal{C}}\rho_{final}P_{\mathcal{C}}) = (1-p)^2 \times 1 + (1-p)\frac{p}{15}(3) + \frac{p}{15}(1-p)(3) + \frac{p^2}{15^2}(9) \]

This expression is identical to the one for $F$.

\[ \mathrm{Tr}(P_{\mathcal{C}}\rho_{final}P_{\mathcal{C}}) = 1 - \frac{8}{5}p + \frac{16}{25}p^2 \]

Finally, the logical state fidelity $F_L$ is:

\[ F_L = \frac{F}{\mathrm{Tr}(P_{\mathcal{C}}\rho_{final}P_{\mathcal{C}})} = \frac{1 - \frac{8}{5}p + \frac{16}{25}p^2}{1 - \frac{8}{5}p + \frac{16}{25}p^2} = 1 \]

This implies that all errors that would change the logical state of $|\psi\rangle$ are detected by the code and are discarded by the post-selection. All errors that do not generate a syndrome (i.e. those that map $|\psi\rangle$ into the code space) are exactly those that are in $\mathcal{S}_\psi$ and thus leave the state $|\psi\rangle$ unchanged. Therefore, with successful post-selection, the logical state fidelity remains 1.

The final answer is $\boxed{1}$.

User #5

Populate your final answer into the code template provided below. This step is purely for formatting/display purposes. No additional reasoning or derivation should be performed. Do not import any modules or packages beyond what is provided in the template.

import sympy as sp

p = sp.symbols('p')

def answer(p):
    r"""
    Return the expression of logical state fidelity of the final 2-qubit logical state
    at the end of the circuit as a function of the two-qubit gate error rate $p$ in Sympy format.

    Inputs
    ----------
    p: sympy.Symbol, the two-qubit gate error rate, $p$

    Outputs
    ----------
    F_logical: sympy.Expr, the logical state fidelity as a function of $p$
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    F_logical = ...  # a SymPy expression of inputs
    # ---------------------------------------------------------------

    return F_logical

Assistant #6

import sympy as sp

p = sp.symbols('p')

def answer(p):
    r"""
    Return the expression of logical state fidelity of the final 2-qubit logical state
    at the end of the circuit as a function of the two-qubit gate error rate $p$ in Sympy format.

    Inputs
    ----------
    p: sympy.Symbol, the two-qubit gate error rate, $p$

    Outputs
    ----------
    F_logical: sympy.Expr, the logical state fidelity as a function of $p$
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    F_logical = 1
    # ---------------------------------------------------------------

    return F_logical