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Efficient quantum algorithms for $GHZ$ and $W$ states, and implementation on the IBM quantum computer
Diogo Cruz, Romain Fournier, Fabien Gremion, Alix Jeannerot, Kenichi Komagata, Tara Tosic, Jarla Thiesbrummel, Chun Lam Chan, Nicolas Macris, Marc-André Dupertuis, Clément Javerzac-Galy
TL;DR
The paper addresses how to generate large multipartite GHZ and W states efficiently for quantum-information applications. It develops logarithmic-step circuits, implements and characterizes them on IBM quantum computers, and reaches 16 qubits, while finding that current decoherence limits quantum error correction and circuit scalability.
Problem
Generating multipartite GHZ_N and W_N states efficiently and at larger sizes remains important for quantum networks and distributed quantum information processing.
Method
The paper develops logarithmic-step GHZ_N and W_N generation algorithms, implements them on IBMQ, and characterizes the states using tomography, parity oscillations, and histogram distance.
Results
The algorithms generated and characterized entangled states with up to 16 superconducting qubits, twice the number previously achieved.
Takeaways & Limitations
The algorithms are potential deterministic building blocks for creating shared GHZ_N or W_N states without prior entanglement, including for arbitrary party numbers.
Takeaways & Limitations
Quantum error correction was limited by current decoherence, while IBM connectivity constraints increased the 16-qubit GHZ circuit from 5 to 10 steps.
Abstract
from arXiv · showhide
We propose efficient algorithms with logarithmic step complexities for the generation of entangled $GHZ_N$ and $W_N$ states useful for quantum networks, and we demonstrate an implementation on the IBM quantum computer up to $N=16$. Improved quality is then investigated using full quantum tomography for low-$N$ GHZ and W states. This is completed by parity oscillations and histogram distance for large $N$ GHZ and W states respectively. We are capable to robustly build states with about twice the number of quantum bits which were previously achieved. Finally we attempt quantum error correction on GHZ using recent schemes proposed in the literature, but with the present amount of decoherence they prove detrimental.
INTRODUCTION
The paper develops logarithmic-step algorithms for generating GHZ_N and W_N states, implements them on IBM quantum computers, and addresses practical connectivity and decoherence constraints. The constructions target scalable entanglement generation for quantum networks, including arbitrary party numbers without prior entanglement.
- Motivation: The paper focuses on GHZ_N and W_N states as representative multipartite entangled resources for distributed quantum information processing.These state classes are distinct under LOCC for N ≥ 3 and are relevant to quantum networks.
- Algorithms: Logarithmic algorithms generate GHZ_N and W_N states in approximately log2 N steps, compared with linear-time constructions.The GHZ construction exploits parallel CNOT operations, while the W construction uses hierarchical dichotomies and controlled rotations.
- GHZ construction: GHZ circuits parallelize CNOT gates by reshuffling control bits while preserving the target GHZ state.For general N, the circuit contains N gates and has approximately log2 N time complexity.
- W construction: W-state circuits use controlled-G(p) rotations followed by inverted CNOTs, with sequential construction requiring approximately N gates and time steps.The basic sequence uses B(1/N), B(1/(N−1)), through B(1/2), while implementations of each block add a constant factor to gate count.
- Implementation constraints: IBM connectivity constraints can increase GHZ circuit depth from 5 to 6 and then to 10 steps for the 16-qubit case.Limited qubit connections require extra CNOT steps, while architectural CNOT-direction restrictions can add further overhead.
CHARACTERIZING GHZN AND WN ON IBMQ
The paper characterizes IBMQ-generated GHZ_N and W_N states using tomography for small N and alternative diagnostics for larger N. Tomography verifies states up to N=5, while parity oscillations and histogram distance extend characterization to N=16, with logarithmic algorithms performing better than linear ones.
- Quantum tomography: Quantum tomography reconstructs the full density matrix but requires 4^N expectation values and many repeated measurements.The method provides maximum information about the state but becomes impractical as N increases.
- Characterization methods: Fidelity exceeded 70% for GHZ_N and W_N up to N = 5, while larger states required parity oscillations, partial tomography, or histogram distance.For GHZ_N, parity-oscillation decay was used up to N = 16; for W_N, partial tomography and histogram distance were used.
- Quantum tomography: Full tomography of a 5-qubit state took about 1 h and would increase by a factor of 4 for each additional qubit.IBMQ access and a limit of approximately 75 parallel circuit evaluations further constrain the workload.
- Reconstructed density matrices: GHZ3 and GHZ4 matrix elements were close to 1/2, while W3 and W4 characteristic values were close to 1/3 and 1/4.The reconstructed matrices also show increasing T1 relaxation toward the ground state and stronger decay of off-diagonal elements with N.
- Algorithm comparison: At N = 4, logarithmic and linear reconstructions look similar, but fidelity and histogram distance distinguish the logarithmic advantage more clearly at larger N.The reconstructed density matrices establish generation of the desired states through GHZ4 and W4.
- Fidelity: The logarithmic algorithm had superior fidelity for both GHZ_N and W_N up to N = 5, with fidelity decaying as N increased.The authors attribute W_N’s slightly lower fidelity to its larger number of steps.
Decay of parity oscillations of GHZN up to large N
For large N, the paper replaces full tomography with parity-oscillation decay to characterize GHZ_N states. This approach measures coherence rather than reconstructing the entire density matrix.
- For large N, GHZ_N characterization uses parity-oscillation decay instead of full tomography.
Principle of measurement
GHZ_N coherence is inferred from parity oscillations generated by rotating every qubit and varying the phase. The oscillation amplitude directly measures the state’s coherence and can be tracked during decoherence.
- GHZ_N coherence is the sum of the magnitudes of the two off-diagonal density-matrix elements connecting |0⟩^⊗N and |1⟩^⊗N.
- Parity is defined as P = P_even − P_odd, where the two probabilities count measured bitstrings with even or odd numbers of 1s.
- Varying the phase φ of rotations applied to each qubit induces parity oscillations whose amplitude directly relates to GHZ_N coherence.
- Decoherence is quantified by measuring coherence across delay times τ between GHZ_N preparation and parity-oscillation measurement.The procedure was applied to entangled systems of up to 16 superconducting qubits.
Results
Parity-oscillation measurements characterize GHZ coherence through N = 16, showing exponential decay with N at zero delay and decoherence rates that increase approximately linearly with N. The logarithmic algorithm extends measurement reach beyond earlier work, while coherence is strongly affected by low-T2 qubits and circuit errors.
- Zero-delay coherence: C(N, 0) decays exponentially with the number of qubits, unlike the linear decay previously observed with a linear-generation method.The paper attributes the difference partly to logarithmic rather than linear state-generation time.
- Delay dependence: Coherence C(N, τ) decreases exponentially with delay time, and exponential fits extract the coherence-time parameter T2(N).The measurements use sinusoidal parity-oscillation fits, with amplitude related to coherence.
- Comparison across N: For N ≤ 12, the generated states have better coherence than the earlier 8-qubit state, while coherences from five qubits onward become closer in value.The convergence may reflect a limiting low-coherence qubit or the increasing number of circuit steps.
- Decoherence scaling: 1/T2(N) scales approximately as 2N, indicating an approximately linear increase in decoherence rate with entangled-qubit number.The measured average decoherence time was Tav ≈ 27 µs, below the single-qubit value of 102.2 µs.
- Decoherence scaling: Tav ≃ 25.45 µs gives a coherence ratio scaling as 0.943N, showing that low-T2 qubits strongly reduce overall state coherence.The authors note that qubits are not equally coherent and that selecting the best qubits first can affect lower-N decay rates.
- Coherence measurements: N = 16 qubits were characterized through parity oscillations, extending the earlier 8-qubit measurement range.The efficient algorithm enabled reliable parity-oscillation measurements up to the IBMQ device limit.
Partial tomography of W(N) for large N values: histogram distance from ideal state populations
Large-N W-state quality is assessed from computational-basis populations and their histogram distance from the ideal distribution. The logarithmic algorithm performs better than the linear algorithm, but quality worsens beyond N = 12 as relaxation increasingly favors the ground state.
- Evaluation at large N: W_N states generated logarithmically reach N = 16, whereas the linear algorithm reaches only N = 9.Full tomography is impractical at these sizes, so evaluation is restricted to computational-basis populations.
- Population analysis: Until N = 12, target 1/N populations clearly dominate unwanted populations, indicating fairly good W_N states.The principal exception is the ground-state component, whose population grows rapidly with N.
- Error mechanism: Relaxation favors the final ground state because longer state creation increases the time available for relaxation, especially in the linear algorithm.The authors connect the growing algorithmic difference with the larger number of steps in linear W_N generation.
- Histogram distance: D_K(N) compares measured and ideal populations, approaching 0 for highly identical distributions and 1 for unrelated distributions.The metric is used because it is easy to compute without full tomography.
- Metric limitation: D_K(N) is not sufficient to directly assert small trace distance, because trace distance upper-bounds Kolmogorov distance rather than conversely.The two distances coincide when the ideal and measured density matrices commute, and approximately coincide for close-to-ideal states.
- Algorithm comparison: For N > 12, histogram distance indicates poor fidelity, while the logarithmic algorithm is clearly superior to the linear algorithm at the same N.The distance rises fairly linearly with N for both algorithms before saturating at unity.
AN ATTEMPT AT QUANTUM ERROR CORRECTION
The study implements a GHZ error-correction scheme using ancillas to store phase and parity information. It tests only three- and four-qubit GHZ states under the IBMQ architecture constraints.
- Correction scheme: The implemented scheme corrects arbitrary phase, phase-flip, and bit-flip errors using ancilla qubits that store GHZ phase and parity information.The cited scheme uses one ancilla for phase-flip correction and two ancillas for bit-flip correction.
- Implementation scope: N = 3 and N = 4 GHZ states were tested, using the linear circuit because the logarithmic circuit offers little benefit at these sizes.For N = 3, the linear and logarithmic circuits are the same.
- Hardware constraints: The implementation respects the 16-qubit processor connectivity, including directed CNOT constraints that may require backend SWAP operations.The GHZ and ancilla qubits are assigned to explicit processor-qubit arrays for the three- and four-qubit tests.
- Hardware constraints: The processor architecture used for the correction experiment is the 16-qubit Rueschlikon device.The architecture is presented in Figure 19.
Construction of ancilla qubits
Ancilla construction uses CNOT and SWAP sequences to gather phase and pairwise parity information from GHZ qubits. After the circuits, one ancilla stores the GHZ phase and the others store adjacent-qubit parities.
- Phase ancilla: A phase ancilla is moved through the GHZ register using repeated CNOT and SWAP operations.The loop shifts the phase ancilla initially at LP[n − 1] to LP[0].
- Parity ancillas: The parity circuit uses paired CNOT and SWAP operations to create ancillas for neighboring GHZ-qubit pairs.For each k, LP[k] stores the parity of LQ[k] and LQ[k + 1].
- Stored syndromes: After construction, LP[N − 1] stores the GHZ phase while LP[0] through LP[N − 2] store pairwise parities.These ancilla values provide the syndrome information used by the correction procedure.
Circuits for error correction
The paper implements previously proposed circuits for arbitrary phase, phase-flip, and bit-flip error correction, including a CNOT- and SWAP-based bit-flip procedure.
- The error-correction circuits follow published schemes for arbitrary phase, phase-flip, and bit-flip correction.The arbitrary-phase and phase-flip circuits use procedures analogous to phase checking.
- The implementation distinguishes the bit-flip procedure from the phase-correction procedures through its explicit gate sequence.The paper describes the bit-flip construction after introducing shared procedures for the other correction circuits.
- The bit-flip circuit applies CNOT gates and SWAP operations iteratively across logical qubits.Its listed sequence alternates CNOT and SWAP operations before continuing to the next index.
Experimental results
The experiments use tomography to compare GHZ3 states with and without error correction, while figure coloring represents density-matrix values. Error correction generally worsens the reconstructed states under the tested conditions.
- Tomography experiment: Tomography evaluates GHZ3 states with and without bit-flip, phase-flip, and arbitrary phase correction.Each density-matrix coefficient was sampled using 1024 shots, without added delays between preparation, correction, and tomography.
- Error-correction outcome: The reconstructed GHZ states generally decay after all three error-correction procedures.The states also generally relax toward |000⟩⟨000|, except for random phase correction in Fig.20(d).
- Tomography experiment: Yellow regions in the tomography figures indicate higher values, while darker regions indicate lower values.This color encoding applies to the reported tomography results.
- Error-correction outcome: A delay comparable to the error-correction circuit produces preliminary evidence of less decoherence than applying error correction.The paper attributes the result to experimental noise exceeding the proposed scheme’s error-correction capability rather than to the scheme itself.
CONCLUSION
The paper proposes and implements logarithmic-step algorithms for generating GHZ_N and W_N states on IBMQ, emphasizing their potential use in quantum networks. Its error-correction tests remain limited by decoherence, while larger and better-connected processors are identified as a route for further demonstrations.
- CONCLUSION: The paper proposes and implements logarithmic-step algorithms for generating entangled GHZ_N and W_N states on IBMQ.The algorithms are presented as deterministic methods for creating shared states without prior entanglement, including for arbitrary numbers of parties.
- CONCLUSION: The algorithms are relevant to quantum networks because they can potentially create shared GHZ_N or W_N states by local extension without prior entanglement.The conclusion identifies applications in distributed quantum information processing.
- CONCLUSION: Quantum error correction on the generated GHZ states is limited by the current level of decoherence.The paper reports a linear increase in decoherence rate with the number of qubits and anticipates better demonstrations with improved processors.