Q_info

 Python changes and Qiskit changes; not always clear where we are. 

got inspired by a qiskit tutorial by Diego Emilio Serrano:

https://youtu.be/ZItpTvwz4cc?si=cnSsK_tJCAI_ZFcq

A sequence that currently works on my Windows installation:

                                                                                   
                                                                                

Compare

 qiskit; how is a density matrix more useful than a statevector representation of state?

Great question! In quantum computing, both density matrices and statevectors are used to represent quantum states, but they serve different purposes and have distinct advantages.

Statevector Representation:

A statevector is a pure state representation of a quantum system. It describes the exact state of a quantum system using a vector in a complex vector space.

It's useful for simulating ideal, noise-free quantum systems where the state is well-defined and isolated.

Density Matrix Representation:

A density matrix is a more general representation that can describe both pure states and mixed states. Mixed states occur when there is uncertainty or decoherence in the system.

It is particularly useful for representing quantum systems that interact with their environment, leading to noise and decoherence.

Density matrices can capture the statistical properties of ensembles of quantum states, making them essential for studying open quantum systems and quantum noise.

In summary, while statevectors are ideal for pure, isolated quantum systems, density matrices provide a more comprehensive and realistic representation of quantum states, especially in the presence of noise and decoherence

                                                                              

Why am I getting dims=(2, 2) when the matrix is a 4x4...

The dims=(2, 2) output indicates that the density matrix is being interpreted as a 2-qubit system. Each qubit has a dimension of 2, so the total dimension of the system is $2\times 2=\mathrm{4, }2\; times\; 2\; =\; 4$, which corresponds to the 4x4 matrix you are seeing.

In Qiskit, the dims parameter specifies the dimensions of the subsystems. For a 2-qubit system, each qubit is a 2-dimensional subsystem, hence dims=(2, 2). This is why you see a 4x4 matrix, as it represents the combined state of the two 2-dimensional qubits.

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My results on the Episode5 code are in!

The results on the tutorial video:

                                                                 

This shows, as expected, that the further the qubit from the original 'on' qubit, the

less likely a true entangled result of 00 or 11...

My results (!?):

                                                                        

Had a chat with Merlin (ChatGPT3.5):

Qiskit; this code endeavours to show gradual decoherence as the number of target qubits increases. Can you explain to me why it is important to keep track of parity?…

In quantum computing, especially in the context of digital quantum circuits, tracking parity is significant for several reasons, particularly in the context of your Qiskit example that demonstrates gradual decoherence as the number of target qubits increases. Let’s explore the importance of tracking parity within this framework.

1. Error Detection and Correction:

Quantum systems are prone to various types of errors, including bit-flip errors (X errors) and phase-flip errors (Z errors). By tracking the parity of the measured ancilla qubits, you can detect changes in the state that indicate whether the logical operation has been successful. Parity checks help to determine if the quantum state has been altered due to decoherence or other errors during computation. This is particularly useful in implementing error correction codes, as the information about parity can help identify which qubits might need to be corrected.

2. Conditional Operations:

The code follows a conditional approach based on the measured parity of the ancilla qubits to apply specific gates (Z and X gates) on the control qubit and the target qubit, respectively. Specifically, if the parity condition is met, a Z-gate or an X-gate is applied. This is akin to creating a feedback mechanism where the results of intermediate qubit measurements (anchored in parity) dictate further operations in the circuit, enabling more controlled manipulations based on the system's state.

3. Entanglement and Correlations:

Understanding parity is also crucial when dealing with entangled qubits. The parity of a set of qubits provides insight into the correlations between them. For example, in your circuit, the creation of Bell pairs and the subsequent measurements on ancilla qubits involve calculating parity to analyze the resultant state of entangled qubits. Parity can reveal whether the overall system is in an even or odd superposition, affecting how the qubits behave collectively.

4. Scalability in Quantum Systems:

As the number of qubits increases, the complexity of quantum circuits grows exponentially. Keeping track of the parity of groups of qubits allows for a more manageable way to scale up quantum circuits without losing track of coherent operations. You can use parity to manage larger systems and mitigate decoherence effects that arise as more qubits are added to the quantum register.

5. Measurement Outcomes:

Lastly, tracking parity can be vital for interpreting the results of measurements in a multi-qubit setting. The parity of the set of measured ancillas can provide a succinct summary of the outcome of several measurements, thus allowing you to draw conclusions about the behavior of the entire quantum state without needing to evaluate each qubit separately.

Summary

In the provided Qiskit code, tracking parity plays a pivotal role in ensuring error detection, implementing conditional logic based on the state of the qubits, assessing entanglement among qubits, and managing the coherence as the system scales with additional qubits. Parity serves as a powerful mechanism to navigate the complexities of quantum computation, especially in a controlled decoherence setting, where maintaining quantum coherence is challenging but necessary for reliable calculations.

To what specific end is parity examined in this code; and are any adjustments made in response?

In the provided Qiskit code, parity is examined specifically to control whether certain gates (Z and X gates) are applied to the control qubit (the first qubit) and the target qubit (the last qubit) based on the measurement results of the ancilla qubits in the circuit. Here’s a detailed breakdown of how parity is utilized and what adjustments are made in response:

Examining Parity

Measurement of Ancilla Qubits:

The code measures several ancilla qubits and keeps track of the parity of these measurements. There are two distinct sets of ancilla qubits:

cr2 (for the Z gate): This classical register tracks the parity of the measurements on staggered ancilla qubits starting with the first one.

cr1 (for the X gate): This classical register tracks the parity of measurements on alternating ancilla qubits starting with the second one.

Calculating Parity:

The code uses bitwise XOR operations to compute the parity of the measured results. For instance:

python

Copy

parity_control = expr.bit_xor(cr2[i-1], parity_control)  # For controlling Z gate

parity_target = expr.bit_xor(cr1[i], parity_target)      # For controlling X gate

This means that if the number of 1s (or true measurements) is even, the parity will be 0 (even parity), and if odd, parity will be 1 (odd parity).

Adjustments Based on Parity

After the parity calculations, the code responds to the parity values by conditionally applying quantum gates:

Conditional Z Gate:

The Z gate is applied to the control qubit if the parity calculated from cr2 (associated with measuring the ancilla qubits) is true (odd parity):

python

Copy

with qc.if_test(parity_control):

    qc.z(0)

This means if the parity of the associated ancilla qubit measurements results in an odd number of 1s, a Z gate will be applied to the control qubit.

Conditional X Gate:

The X gate is applied to the target qubit if the parity calculated from cr1 (also associated with measuring the ancilla qubits) is true (odd parity):

python

Copy

with qc.if_test(parity_target):

    qc.x(-1)

If the parity derived from the measurements of the associated ancilla qubits is odd, then an X gate will be applied to the last qubit (the target).

Summary of Purpose and Adjustments

Purpose: The examination of parity in this code serves two main purposes:

Conditionally Adjusting Gates: The code uses the parity of specific groups of ancilla qubits to decide whether to apply corrective operations (Z or X gates) on critical qubits. This interaction helps mitigate potential errors and modifies the state based on the outcomes of earlier measurements.

Maintaining Quantum Coherence: By adjusting the gates based on parity, the code aims to manage the effects of decoherence that may arise as qubits become more coupled in the quantum system, enabling better control over the quantum state.

Adjustments Made: The adjustments involve the conditional application of gates based on the outcome of the parity checks, effectively controlling the state of the qubits based on the measurements of the ancilla qubits. This mechanism is essential for maintaining the desired behavior of the overall circuit as qubits are added, and decoherence is considered.

So there already are measures to mitigate decoherence in the code…

Yes, the code demonstrates some measures that can help mitigate the effects of decoherence, though it's important to note that the code primarily focuses on monitoring and responding to quantum states rather than implementing full-fledged error correction. Here’s how the structure of the code contributes to managing decoherence:

Measures to Mitigate Decoherence

Use of Ancilla Qubits:

Ancilla Qubits for Error Detection: The code employs ancilla qubits to check the parity of other qubits in the system. By measuring these ancillary qubits, the circuit can detect if errors have occurred in the main computational qubits. This is important because it allows the quantum circuit to assess the state of the system without directly inferring from the main qubits that might be corrupted due to decoherence.

Conditional Gates Based on Measurements:

Dynamic Response to Errors: The conditional application of the Z and X gates based on the measured parities of the ancilla qubits acts as a reactive measure. If the parity indicates that decoherence has affected the control or target qubits, appropriate corrective gates are applied to adjust their states:

The Z-gate modifies the phase of the control qubit in response to identified errors.

The X-gate flips the state of the target qubit if the corresponding parity check indicates an error.

Parity Checks:

Error Characterization: By maintaining and checking the parity of the ancilla measurements, the circuit can characterize whether the quantum state has shifted due to decoherence. This information can be crucial to understanding how the system behaves under different conditions, including increased coupling or other error-prone environments.

Creating Entangled States:

Entanglement Helps with Coherence: The use of controlled operations (CNOT gates) among the qubits creates entanglement. While entanglement can be sensitive to decoherence, it can also provide a specific structure that might help retain information in the presence of noise.

Limitations and Considerations

While the measures in the code help detect and react to potential issues arising from decoherence, they may not be sufficient for full error correction or fault tolerance:

No Full Error Correction Scheme: The code does not implement a comprehensive error correction code such as Shor's code or the surface code, which would provide more robust protection against various types of errors, including depolarizing noise and other forms of decoherence.

Classical Post-Processing: The protocol relies on classical measurements to decide which quantum operations to perform after the fact — this is not inherently real-time feedback, which is often sought in active error correction schemes.

Summary

Overall, while the code implements some mechanisms aimed at managing the impact of decoherence and errors, it primarily demonstrates a foundational level of error detection and correction strategy. As quantum circuits become more complex and include additional qubits, more sophisticated error correction protocols will likely be required to maintain coherence and ensure reliable computation. This is an essential area of research as quantum computing systems scale up.

Episode 5

 I am currently running the code from the Coding with Qiskit Episode 5 tutorial. Should

see results by tomorrow morning. I am on ibm_brisbane because ibm_sherbrooke is on

maintenance.

Asked Merlin (ChatGPT4o) for a description of the set-up:

The provided Qiskit code defines a function `get_dynamic_CNOT_circuit(num_qubit)` that creates a quantum circuit with the purpose of implementing a dynamic CNOT operation over a 1D chain of nearest-neighbor qubits. Let's break down the setup and reasoning behind the circuit:

1. Initialization of Registers and Circuit:

   - A quantum register `qr` of size `num_qubit` is created. This includes:

     - The 0th qubit as the control.

     - The last qubit (`num_qubit - 1`) as the target.

     - Remaining qubits serve as ancillae for facilitating the operation.

   - Three classical registers are used:

     - `cr1` to store parity results for conditional X gates.

     - `cr2` to store parity results for conditional Z gates.

     - `cr3` for the final measurement results of the control and target qubits.

2. Preparation of the Control Qubit:

   - The control qubit (0th qubit) is initialized to the `+` state using a Hadamard gate (`qc.h(0)`).

3. **Entangling Control and First Ancilla:**

   - A CNOT gate entangles the control qubit with the first ancilla qubit (`qc.cx(0, 1)`).

4. Creation of Bell Pairs:

   - Bell pairs are created among ancilla qubits to facilitate interaction across the chain:

     - A series of Hadamard (`qc.h`) and CNOT gates (`qc.cx`) are applied on pairs of ancilla qubits.

5. Staggered Bell State Preparation:

   - Additional entanglement is created between alternating qubits, including the target qubit, using CNOT gates.

   - Another set of Hadamard gates is applied to complete these entanglements.

6. Measurement and Conditional Gates:

   - Measurements are performed on alternating ancilla qubits, storing results in classical registers `cr1` and `cr2`.

   - These measurement results are used to determine parities for conditional quantum gates:

     - The parity from `cr2` is used to apply a conditional Z gate on the control qubit.

     - The parity from `cr1` is used to apply a conditional X gate on the target qubit.

7. Final Measurements:

   - The state of the control qubit (0th qubit) and the target qubit are measured, with results recorded in `cr3`.

Reasoning:

The circuit essentially implements a non-standard CNOT operation where the control and target qubits are not adjacent but connected via a series of ancillae. The Bell pairs and conditional gates ensure that the operation can be translated across these ancillae, making use of the nearest-neighbor connectivity paradigm. This dynamic CNOT approach is useful for systems where long-range interactions need to be simulated via short-range entanglements and classical feedback.

What happens next; are there only 7 qubits for illustration, but actually 41 in the circuit?

In the provided code snippet, the purpose is to generate dynamic CNOT circuits with increasing numbers of qubits, starting from 7 qubits and going up to a maximum of 41 qubits (inclusive). Here's a breakdown of what happens next:

 Explanation of the Code

1. Initial Circuit Creation:

   - `qc = get_dynamic_CNOT_circuit(num_qubit=7)` creates a quantum circuit with exactly 7 qubits. This specific case serves as an illustration of how the circuit is formed with fewer qubits.

2. Setup for Multiple Circuits:

   - `max_num_qubit = 41` defines the maximum number of qubits.

   - `qc_list = []` initializes an empty list to store all the circuits generated.

   - `num_qubit_list = list(range(7, max_num_qubit + 1, 2))` creates a list of odd numbers starting from 7 up to 41 (inclusive). This means the circuits generated will have 7, 9, 11, ..., 41 qubits.

3. Loop to Generate Circuits:

   - `for num_qubit in num_qubit_list:` iterates through each number of qubits in `num_qubit_list`.

   - `qc_list.append(get_dynamic_CNOT_circuit(num_qubit))` calls the function to create a dynamic CNOT circuit for the current `num_qubit` and appends it to `qc_list`.

 Purpose of the Circuit Creation

- The function `get_dynamic_CNOT_circuit(num_qubit)` you've already looked at is called iteratively for each odd number of qubits from 7 to 41.

- This creates a series of circuits that demonstrate how the dynamic CNOT gate can be implemented for increasing qubit sizes, potentially useful for studying how the circuit scales and behaves as the number of qubits increases.

 Summary

Yes, while the initial example uses 7 qubits for demonstration purposes, the overall structure of the code is designed to generate quantum circuits with a variety of sizes, ending with 41 qubits. This allows for experimentation with various configurations, tests of quantum algorithms, and a better understanding of how circuit depth and complexity evolve with more qubits. 

What is the GHZ state?

                                                                         

                                                                                    

*     *     *

Asked Perplexity Ai to summarize the paper on which this circuit is based:

Answer

The paper titled "Efficient Long-Range Entanglement using Dynamic Circuits" explores innovative methods for generating long-range entanglement in quantum systems through the use of dynamic circuits, which incorporate non-unitary operations and measurements.

Key Highlights:

Dynamic Circuits: The authors propose that dynamic circuits can surpass traditional unitary dynamics in efficiency when creating intricate entangled states. These circuits utilize measurements and conditional feed-forward operations, allowing for enhanced connectivity and state preparation in quantum devices.

Main Tasks:

CNOT Gate Teleportation: The study demonstrates the teleportation of a CNOT gate across up to 101 qubits by utilizing 99 mid-circuit measurement outcomes. This approach shows improved performance compared to conventional unitary methods.

GHZ State Preparation: The authors also focus on preparing Greenberger-Horne-Zeilinger (GHZ) states, providing a detailed error budget that outlines the challenges that need to be addressed to fully leverage dynamic circuits.

Future Implications: The findings suggest that dynamic circuits could play a crucial role in achieving long-range entanglement on large-scale quantum devices in the near future, potentially unlocking new capabilities in quantum computing.

This research indicates significant advancements in the field of quantum simulation and entanglement generation, highlighting the potential of dynamic circuits for practical applications in quantum technology.

Tell me more about the error budget.

Answer

The error budget in quantum computing refers to the detailed analysis and accounting of various error sources that can affect the performance of quantum systems. This concept is critical for understanding how different components contribute to overall logical error rates, especially in the context of quantum error correction (QEC).

Key Aspects of Error Budgets in Quantum Computing:

Components of Error Budgets:

Physical Errors: These include errors arising from gate operations, decoherence, and measurement inaccuracies. Each type of physical error has a distinct impact on the logical qubit's performance.

Error Suppression Factor: This factor quantifies how much the logical error rate can be reduced by increasing the size or distance of the error-correcting code. For instance, a higher distance code generally leads to better error suppression.

Dominant Error Sources:

Research shows that specific types of errors, such as controlled-Z (CZ) gate errors and data qubit decoherence during measurement, are often the most significant contributors to logical errors in quantum circuits. Understanding these dominant sources allows researchers to prioritize improvements in those areas.

Performance Thresholds:

Quantum error correction becomes effective only when the physical error rates are below a certain threshold. The error budget helps determine what improvements are necessary to achieve this threshold and thus make quantum computations feasible.

Estimation and Projections:

By simulating various scenarios with different physical error rates, researchers can estimate how much improvement is needed across different components to achieve desired logical performance levels. For example, projections may indicate that a 20% improvement in component performance is required to move below the error threshold.

Experimental Validation:

While theoretical models provide a framework for understanding error budgets, experimental validation is crucial. Researchers conduct tests with larger codes and longer durations to confirm that their projections hold true in practical scenarios.

Future Directions:

As quantum technologies evolve, continuous refinement of the error budget will be necessary to address new challenges and improve scalability. This includes developing more sophisticated QEC codes and optimizing existing quantum architectures.

In summary, an effective error budget is essential for advancing quantum computing technologies by identifying critical areas for improvement and guiding research efforts towards achieving fault-tolerant quantum systems.

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https://youtu.be/QDPtcwhpQkE?si=AHd_WSbvccYQbDk8

Semi_D

                                                                         

                                                                               

                                                              *     *     *

                                                                         

Cool

COPILOT: 

Qiskit; what is an operator 'op'?

                                                                  

Operators module overview | IBM Quantum Documentation

Are op and PauliOp both found in Quantum Physics; what do they do?

                                                                    

                                                                                          

Tagtik

 From Belgian news agency Tagtik:

"La Russie et les Etats-Unis sont au bord d'un conflit militaire direct"

                                                 *     *     *

La Pologne construit un rideau de fer à sa frontière

I am not sure how to view this Iron Curtain II development. with Iron Curtain I, 

the then USSR built up defenses to keep western influence at bay, trying to shield states 

it was guarding. Are we headed with the same situation where the West tries to shield those 

same states from the East!?

Iron Curtain - Wikipedia

PauliVec

                                                                                

META AI:

Qiskit PauliVec; I have a 2-qubit circuit with the first a Hadamard gate and the second a cx with q(0) as control and q(1) as target. A PauliVec representation shows the YY as negative1, while the other three are positive. What accounts for this?

A nice question about Qiskit and quantum computing!

The PauliVec representation you're referring to is likely the Pauli Vector representation of the density matrix of your 2-qubit circuit. This representation decomposes the density matrix into a linear combination of Pauli matrices.

In your case, the circuit consists of:

Hadamard gate (H) on qubit 0

Controlled-NOT gate (CX) with qubit 0 as control and qubit 1 as target

To understand why the YY term is negative, let's analyze the circuit:

Step 1: Hadamard gate on qubit 0

The Hadamard gate creates a superposition state on qubit 0: |0+ |1. This can be represented as:

ρ = |ψψ| = (|00+ |10+ |01+ |11) / 2

Step 2: Controlled-NOT gate

The CX gate applies a NOT operation to qubit 1 if qubit 0 is in the state |1. This creates entanglement between the qubits.

After applying CX, the state becomes:

ρ = |ψψ| = (|00+ |11- |01+ |10) / 2

Notice the relative phase difference between |00and |11.

Pauli Vector representation

Now, let's decompose the density matrix ρ into Pauli matrices:

ρ = (I ⊗ I + X ⊗ X + Y ⊗ Y - Z ⊗ Z) / 4

Here:

I is the identity matrix

X, Y, Z are the Pauli matrices

⊗ denotes the tensor product

The coefficients of the Pauli Vector representation are:

II: +1

XX: +1

YY: -1 (this is what you observed)

ZZ: +1

The negative coefficient for YY arises from the relative phase difference between |00and |11in the entangled state. Specifically, it's due to the antisymmetric nature of the Bell state |ψ= (|00- |11)/√2, which is present in the density matrix.

In summary, the negative YY term in the PauliVec representation is a consequence of the entanglement generated by the CX gate and the relative phase difference between the computational basis states.

Could I see a calculation of the tensor product for the two Ys?

Here's the calculation for the tensor product of two Y Pauli matrices:

Y ⊗ Y =

= |0⊗0 |0⊗1 |1⊗0 |1⊗1

|0 -i 0 0 | |0 0 -i 0 |

|0 i 0 0 | |0 0 i 0 |

|0 0 -i 0 | |0 -i 0 0 |

|0 0 i 0 | |0 0 0 -i |

Now, let's calculate the expectation value of Y ⊗ Y in the state:

ρ = (|00+ |11- |01+ |10) / 2

We'll compute the trace of the product ρ(Y ⊗ Y):

Tr[ρ(Y ⊗ Y)] =

= (∑ij ρij (Y ⊗ Y)ji) / 2

= [ρ00 (Y ⊗ Y)00 + ρ11 (Y ⊗ Y)11 + ρ01 (Y ⊗ Y)10 + ρ10 (Y ⊗ Y)01] / 2

Using ρij values from ρ:

ρ00 = ρ11 = 1/2, ρ01 = ρ10 = 0

Tr[ρ(Y ⊗ Y)] =

= [(1/2)(-i) + (1/2)(i)] / 2

= -1/2 - 1/2

= -1

So, the coefficient of YY in the PauliVec representation is indeed -1

PERPLEXITY:

                                                                         

                                                                                  

COPILOT:

                                                                                   

                                                                ...