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+#
+# Linear GHZ circuit +#

Linear GHZ circuit

+#
+#
+# Log-depth GHZ circuit +#

Log-depth GHZ circuit

+#
+#
+# +# The GHZ state can be prepared using a sequence of Hadamard and CNOT gates. In a linear (sequential) implementation, the CNOT gates are applied one after another, resulting in a circuit depth that grows linearly with the number of qubits. In contrast, a log-depth (parallel) implementation arranges the CNOT gates so that multiple gates acting on disjoint qubits can execute simultaneously, reducing the overall depth to logarithmic in the number of qubits. +# %% +import cirq +import matplotlib.pyplot as plt +import numpy as np +from bloqade import squin, cirq_utils +import bloqade.cirq_utils as utils +from cirq.contrib.svg import SVGCircuit + + +# %% +def build_linear_ghz(n_qubits: int) -> cirq.Circuit: + """Build linear GHZ circuit using squin and convert to Cirq.""" + + @squin.kernel + def linear_ghz_kernel(): + q = squin.qalloc(n_qubits) + squin.h(q[0]) + for i in range(n_qubits - 1): + squin.cx(q[i], q[i + 1]) + + # Create LineQubits for compatibility with existing code + qubits = cirq.LineQubit.range(n_qubits) + circuit = cirq_utils.emit_circuit(linear_ghz_kernel, circuit_qubits=qubits) + return circuit + + +def build_log_ghz(n_qubits: int) -> cirq.Circuit: + """Build logarithmic-depth GHZ circuit using squin and convert to Cirq.""" + import math + + max_iterations = math.ceil(math.log2(n_qubits)) if n_qubits > 1 else 1 + + @squin.kernel + def log_ghz_kernel(): + q = squin.qalloc(n_qubits) + squin.h(q[0]) + + for level in range(max_iterations): + width = 2**level + for i in range(n_qubits): + if i < width: + target = i + width + if target < n_qubits: + squin.cx(q[i], q[target]) + + # Create LineQubits for compatibility with existing code + qubits = cirq.LineQubit.range(n_qubits) + circuit = cirq_utils.emit_circuit(log_ghz_kernel, circuit_qubits=qubits) + return circuit + + +# %% [markdown] +# ### The benefits of parallelism +# We'll run noise simulations for both circuits and compare their fidelities as we scale the number of qubits. +# +# See our blog post [Simulating noisy circuits for near-term quantum hardware](https://bloqade.quera.com/latest/blog/2025/07/30/simulating-noisy-circuits-for-near-term-quantum-hardware/) for detailed information about the noise model used here. The analysis workflow is: +# +# 1. Build a noiseless (ideal) circuit. +# 2. Choose a noise model (we use the Gemini noise model). +# 3. Apply the noise model to the circuit to produce a noisy circuit. +# 4. Simulate the noisy circuit to obtain the final density matrix. +# 5. Simulate the ideal circuit and compare its state to the noisy density matrix to compute fidelity. +# + +# %% + +# Initialize noise model (using Gemini one-zone architecture) +noise_model = utils.noise.GeminiOneZoneNoiseModel() +simulator = cirq.DensityMatrixSimulator() + +# Initialize lists to store fidelities +fidelities_linear = [] +fidelities_log = [] + +# %% [markdown] +# We run noise-model simulations for circuit sizes from 3 to 9 qubits and compute the fidelity (the higher is better). The ideal noiseless circuit has fidelity 1 by construction. + +# %% +qubits = range(3, 9) +# Test both linear and log GHZ circuits with noise model +for n in qubits: + # Linear GHZ circuit + linear_circuit = build_linear_ghz(n) + + # Log GHZ circuit + log_circuit = build_log_ghz(n) + + # Apply noise model + linear_noisy_circuit = utils.noise.transform_circuit( + linear_circuit, model=noise_model + ) + log_noisy_circuit = utils.noise.transform_circuit(log_circuit, model=noise_model) + + # Simulate noiseless circuits + rho_linear = simulator.simulate(linear_circuit).final_density_matrix + rho_log = simulator.simulate(log_circuit).final_density_matrix + + # Simulate noisy circuits + rho_linear_noisy = simulator.simulate(linear_noisy_circuit).final_density_matrix + rho_log_noisy = simulator.simulate(log_noisy_circuit).final_density_matrix + + # Calculate fidelities + fidelity_linear = np.trace(rho_linear @ rho_linear_noisy).real + fidelity_log = np.trace(rho_log @ rho_log_noisy).real + + # Store results + fidelities_linear.append(fidelity_linear) + fidelities_log.append(fidelity_log) + + print(f"n={n}:") + print(f" Linear GHZ: {fidelity_linear:.4f}") + print(f" Log GHZ: {fidelity_log:.4f}") + + +# %% [markdown] +# Fidelity comparison plot: + +# %% +# Create comparison plot +plt.figure(figsize=(10, 6)) + +plt.plot( + qubits, + fidelities_linear, + "ro-", + label="Linear GHZ", + linewidth=2, + markersize=8, +) +plt.plot( + qubits, + fidelities_log, + "bo-", + label="Log-depth GHZ", + linewidth=2, + markersize=8, +) + +plt.xlabel("Number of Qubits", fontsize=14) +plt.ylabel("Fidelity", fontsize=14) +plt.title( + "GHZ State Fidelity Comparison: Linear vs Log-Depth Circuits", + fontsize=16, +) +plt.legend(fontsize=12) +plt.grid(True, alpha=0.3) +plt.xticks(qubits) + +# Add annotations for better understanding +plt.text( + 0.02, + 0.98, + "Higher fidelity = Better performance", + transform=plt.gca().transAxes, + fontsize=12, + verticalalignment="top", + bbox=dict(boxstyle="round", facecolor="wheat", alpha=0.5), +) + +plt.tight_layout() +plt.show() + +# Print summary statistics +print("\n=== Summary Statistics ===") +print( + f"Linear GHZ: Mean = {np.mean(fidelities_linear):.4f}, Std = {np.std(fidelities_linear):.4f}" +) +print( + f"Log-depth GHZ: Mean = {np.mean(fidelities_log):.4f}, Std = {np.std(fidelities_log):.4f}" +) + + +# %% [markdown] +# The GHZ results show that parallelizing gates increases fidelity compared with the sequential implementation. The log-depth circuit consistently outperforms the linear-depth circuit, with the advantage growing as we increase the number of qubits. + +# %% [markdown] +# ## Automatic toolkits for circuit parallelization +# +# In the GHZ example the log-depth circuit was provided manually. Bloqade also includes automatic tools that compress a deep circuit into a more parallel form by casting the problem as an optimization (integer linear program). +# +# We first transpile the circuit into a standard gate set consisting of single-qubit gates and two-qubit CZ gates. CZ gates acting on disjoint qubits commute and therefore can be placed in the same moment. +# +# The optimization objective encourages gates that can execute in parallel to be assigned to nearby moments. Formally, we minimize an objective such as $\sum_{g}\sum_{o_p,o_q \in g} w_g \left|t_p-t_q\right|$, where each group $g$ contains operations with a shared tag. Within each group, an attraction force of strength $w_g$ is introduced. Here $t_p$ and $t_q$ are integer labels (its epoch) of operators $o_p$ and $o_q$. Each operation (gate) can have multiple tags. +# +# To preserve circuit equivalence, we only reorder operations that commute. Execution dependencies are represented as a directed acyclic graph (DAG); each vertex $i$ receives an integer label $t_i$ (its epoch) and ordering constraints are expressed as inequalities on these labels. These constraints and objective terms are encoded in an integer linear program (ILP). +# +# Because the ILP formulation has totally unimodular constraint structure in our encoding, the relaxed linear program yields integer solutions, which makes the optimization efficient in practice. Absolute-value terms are reformulated into equivalent linear forms during construction. +# +# The helper `bloqade.utils.auto_similarity` tags operations and assigns weights to build the linear objective. Users can also add manual annotations (tags/weights) to guide the parallelizer when they need fine-grained control. +# +# + +# %% [markdown] +# ## Example 2: [7,1,3] Steane code circuit + +# %% [markdown] +# We construct a sequential circuit for the [7,1,3] Steane code and then apply Bloqade's parallelization utilities (which use `auto_similarity`) to compress it. + + +# %% [code] +def build_steane_code_circuit(): + """Build the Steane code circuit for error correction using squin and convert to Cirq.""" + + @squin.kernel + def steane_kernel(): + q = squin.qalloc(7) + + # H gate on qubits 1, 2, 3 + squin.h(q[1]) + squin.h(q[2]) + squin.h(q[3]) + + # Encode the logical qubit with CZ and H gates (equivalent to CNOT) + squin.h(q[0]) + squin.cz(q[1], q[0]) + squin.cz(q[2], q[0]) + squin.h(q[4]) + squin.cz(q[2], q[4]) + squin.cz(q[6], q[4]) + squin.h(q[5]) + squin.cz(q[6], q[5]) + squin.cz(q[3], q[5]) + squin.cz(q[1], q[5]) + squin.h(q[5]) + squin.h(q[6]) + squin.cz(q[1], q[6]) + squin.cz(q[2], q[6]) + squin.h(q[6]) + squin.cz(q[3], q[4]) + squin.h(q[4]) + squin.cz(q[3], q[0]) + squin.h(q[0]) + + # Create LineQubits for compatibility with existing code + qubits = cirq.LineQubit.range(7) + circuit = cirq_utils.emit_circuit(steane_kernel, circuit_qubits=qubits) + return circuit + + +# %% +# Build Steane circuits (reuse already defined noise models and simulator) +steane_original = build_steane_code_circuit() +steane_parallel = utils.parallelize(circuit=steane_original) +steane_parallel = utils.remove_tags(steane_parallel) + +# Display original circuit - renders nicely in Jupyter, shows text in terminal +print("Original Steane Circuit:") +try: + get_ipython() # Check if we're in IPython/Jupyter + # In Jupyter: display as SVG (prettier visualization) + from IPython.display import display + + display(SVGCircuit(steane_original)) +except NameError: + # Not in Jupyter: print text representation + print(steane_original) + +print("\nParallelized Steane Circuit:") +try: + get_ipython() + from IPython.display import display + + display(SVGCircuit(steane_parallel)) +except NameError: + print(steane_parallel) +print("Original Steane circuit depth:", len(steane_original)) +print("Parallelized Steane circuit depth:", len(steane_parallel)) + + +# %% [markdown] +# We perform noise analysis on both the original and parallelized Steane circuits. + +# %% +# Apply noise model to both circuits +steane_original_noisy = utils.noise.transform_circuit( + steane_original, model=noise_model +) +steane_parallel_noisy = utils.noise.transform_circuit( + steane_parallel, model=noise_model +) + +# Simulate ideal circuits +rho_original_ideal = simulator.simulate(steane_original).final_density_matrix +rho_parallel_ideal = simulator.simulate(steane_parallel).final_density_matrix + +# Simulate noisy circuits +rho_original_noisy = simulator.simulate(steane_original_noisy).final_density_matrix +rho_parallel_noisy = simulator.simulate(steane_parallel_noisy).final_density_matrix + +# Calculate fidelities +fidelity_original = np.trace(rho_original_ideal @ rho_original_noisy).real +fidelity_parallel = np.trace(rho_parallel_ideal @ rho_parallel_noisy).real + +# Print results +print("\n=== Steane Code Circuit Fidelity Comparison ===") +print(f"Original circuit: {fidelity_original:.4f}") +print(f"Parallelized circuit: {fidelity_parallel:.4f}") + +# Calculate improvement +improvement = fidelity_parallel - fidelity_original + +print(f"\nFidelity improvement with parallelization: {improvement:+.4f}") + + +# Summary analysis +depth_reduction = len(steane_original) - len(steane_parallel) +depth_reduction_pct = ( + (depth_reduction / len(steane_original) * 100) if len(steane_original) > 0 else 0 +) + +print("\n=== Analysis Summary ===") +print( + f"Circuit depth reduction: {len(steane_original)} → {len(steane_parallel)} moments" +) +print(f"Depth reduction: {depth_reduction} moments ({depth_reduction_pct:.1f}%)") +if improvement > 0: + print("✓ Parallelization improves fidelity") +else: + print("✗ Parallelization does not improve fidelity") + + +# %% [markdown] +# ## Example 3: Linear chained circuit +# +# Here is another example of a circuit of CZ gates in a linear chain. The original circuit has a linearly growing number of moments with the number of qubits. + + +# %% +def build_circuit3(): + """Build a linear CZ chain circuit using squin and convert to Cirq.""" + n = 10 + + @squin.kernel + def cz_chain_kernel(): + q = squin.qalloc(n) + for i in range(n - 1): + squin.cz(q[i], q[i + 1]) + + # Create LineQubits for compatibility with existing code + qubits = cirq.LineQubit.range(n) + circuit = cirq_utils.emit_circuit(cz_chain_kernel, circuit_qubits=qubits) + return circuit + + +# Build the linear CZ circuit +circuit3 = build_circuit3() + +# Display original circuit - renders nicely in Jupyter, shows text in terminal +print("Original CZ chain circuit:") +try: + get_ipython() # Check if we're in IPython/Jupyter + # In Jupyter: display as SVG (prettier visualization) + from IPython.display import display + + display(SVGCircuit(circuit3)) +except NameError: + # Not in Jupyter: print text representation + print(circuit3) + +# Parallelize the circuit, `auto_similarity` is automatically applied inside `parallelize` +circuit3_parallel = utils.parallelize(circuit=circuit3) + +# Remove any tags and print the parallelized circuit +circuit3_parallel = utils.remove_tags(circuit3_parallel) +print("Parallelized CZ chain circuit:") +try: + get_ipython() + from IPython.display import display + + display(SVGCircuit(circuit3_parallel)) +except NameError: + print(circuit3_parallel) + +print("Original CZ chain circuit depth:", len(circuit3)) +print("Parallelized CZ chain circuit depth:", len(circuit3_parallel)) +# %% [markdown] +# ## Example 4: 6D Hypercube Circuit +# +# This example demonstrates parallelization on a 6-dimensional hypercube with 64 qubits arranged in a 4×16 array, which can be implemented on our Gemini machine with (5x17) architecture. +# +# The 6D hypercube (2x2x2x2x2x2) is reshaped into two dimenison (4x16). Each qubit's index maps to (x, y) coordinates: +# - Binary index: [y1][y0][x3][x2][x1][x0] +# - x-coordinate (0-15): lower 4 bits +# - y-coordinate (0-3): upper 2 bits +# +# Each of the 6 dimensions connects qubits that differ by exactly one bit, creating a highly connected quantum circuit with 192 CZ gates across 64 qubits. +# + + +# %% +def build_hypercube(): + """ + Build a 6D hypercube circuit with 64 qubits arranged in a 4x16 array using squin and convert to Cirq. + + The hypercube structure follows the mapping from Task4 analysis: + - Binary index: [y1][y0][x3][x2][x1][x0] + - x-coordinate (0-15): lower 4 bits [x3][x2][x1][x0] + - y-coordinate (0-3): upper 2 bits [y1][y0] + + Each of the 6 dimensions connects qubits that differ by exactly one bit: + - Dimension 0: flip bit 0 (x ± 1) + - Dimension 1: flip bit 1 (x ± 2) + - Dimension 2: flip bit 2 (x ± 4) + - Dimension 3: flip bit 3 (x ± 8) + - Dimension 4: flip bit 4 (y ± 1) + - Dimension 5: flip bit 5 (y ± 2) + """ + # Create a 4x16 array of qubits (64 qubits total) + num_rows = 4 # y-coordinates: 0-3 + num_cols = 16 # x-coordinates: 0-15 + + # Helper function to convert hypercube index to (x, y) + def index_to_coord(idx): + x = idx & 0b001111 # Lower 4 bits + y = (idx >> 4) & 0b11 # Upper 2 bits + return x, y + + # Build the list of gate pairs for all 6 dimensions + gate_pairs = [] + + for dim in range(6): + # For each qubit, connect it to its neighbor along dimension `dim` + for idx in range(64): + # Get the neighbor by flipping bit `dim` + neighbor_idx = idx ^ (1 << dim) + + # Only add each edge once (avoid duplicates) + if neighbor_idx > idx: + # Convert indices to coordinates + x1, y1 = index_to_coord(idx) + x2, y2 = index_to_coord(neighbor_idx) + + # Verify coordinates are valid + if ( + 0 <= x1 < num_cols + and 0 <= y1 < num_rows + and 0 <= x2 < num_cols + and 0 <= y2 < num_rows + ): + gate_pairs.append((y1, x1, y2, x2)) + + # Shuffle the gate pairs to make parallelization more challenging + # Without shuffling, gates are already grouped by dimension which makes + # parallelization trivial. Shuffling forces the optimizer to work harder. + import random + + random.seed(0) # Use seed=0 for optimal parallelization (depth 6) + random.shuffle(gate_pairs) + + @squin.kernel + def hypercube_kernel(): + q = squin.qalloc(num_rows * num_cols) + for pair in gate_pairs: + # Convert 2D indices to 1D: index = y * num_cols + x + idx1 = pair[0] * num_cols + pair[1] + idx2 = pair[2] * num_cols + pair[3] + squin.cz(q[idx1], q[idx2]) + + # Create GridQubits for 4x16 array (returns a flat list of 64 qubits) + qubits = cirq.GridQubit.rect(num_rows, num_cols) + circuit = cirq_utils.emit_circuit(hypercube_kernel, circuit_qubits=qubits) + return circuit + + +# %% +# Build and analyze the hypercube circuit +hypercube_circuit = build_hypercube() +print("Original hypercube circuit:") +print("Number of qubits: 64 (arranged in 4×16 array)") +print(f"Number of gates: {len(list(hypercube_circuit.all_operations()))}") +print(f"Circuit depth: {len(hypercube_circuit)} moments") + +# Calculate number of edges per dimension +# Each dimension should have 32 edges (64 qubits / 2) +print("\nExpected edges per dimension: 32") +print("Total edges across 6 dimensions: 192") + +# %% [markdown] +# The original hypercube circuit has 192 CZ gates (32 edges per dimension × 6 dimensions). +# The gates are shuffled randomly to make parallelization more challenging - without shuffling, +# gates would already be grouped by dimension, making the problem trivial. After shuffling, +# Cirq's default sequential insertion creates a deeper circuit, providing a better benchmark +# for the parallelization optimizer. +# +# Visualization is generated separately. The 4×16 grid layout shows: +# - Straight lines for nearest-neighbor interactions (Euclidean distance = 1) +# - Curved arcs for longer-range interactions (distance > 1) to avoid overlapping +# +# ![Original Hypercube Circuit](figures/hypercube_original.png) +# +# The links with same color belongs to the same moment are executed symmataniously within one layer. + +# %% [markdown] +# Now we try to parallel the hypbercube circuit. + +# %% +# Parallelize the hypercube circuit. There're some randomness in the optimizer which leads to different results in different runs. +print("\nParallelizing hypercube circuit...") +hypercube_parallel = utils.parallelize(circuit=hypercube_circuit) +hypercube_parallel = utils.remove_tags(hypercube_parallel) + +print(f"\nOriginal hypercube circuit depth: {len(hypercube_circuit)}") +print(f"Parallelized hypercube circuit depth: {len(hypercube_parallel)}") +depth_reduction = len(hypercube_circuit) - len(hypercube_parallel) +depth_reduction_pct = depth_reduction / len(hypercube_circuit) * 100 +print(f"Depth reduction: {depth_reduction} moments ({depth_reduction_pct:.1f}%)") + +# %% [markdown] +# The parallelization significantly reduces the circuit depth by identifying gates that act on +# disjoint qubits and can execute simultaneously. The hypercube's regular structure allows many +# gates to be grouped into the same moment. +# +# Visualization of the parallelized circuit shows gates colored by their assigned moment: +# +# ![Parallelized Hypercube Circuit](figures/hypercube_parallel.png) +# +# The parallelized circuit demonstrates that quantum algorithms on hypercube connectivity can +# benefit substantially from automatic parallelization, potentially improving fidelity by +# reducing overall execution time and exposure to decoherence. diff --git a/docs/digital/tutorials/circuits_with_bloqade.py b/docs/digital/tutorials/circuits_with_bloqade.py index fa4e8446a..4152fcb05 100644 --- a/docs/digital/tutorials/circuits_with_bloqade.py +++ b/docs/digital/tutorials/circuits_with_bloqade.py @@ -21,6 +21,7 @@ Register = IList[Qubit, Any] + # %% @squin.kernel def hello_world(theta: float) -> IList[MeasurementResult, Any]: diff --git a/docs/digital/tutorials/figures/ghz_linear_circuit.svg 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