Quantum Computing’s Entanglement Costs Finally Quantified for Key Operations

Scientists are increasingly focused on quantifying the resources required for non-local quantum computation, a field with implications spanning complexity theory, cryptography and even gravity. Richard Cleve of the Institute for Quantum Computing and School of Computer Science, University of Waterloo, alongside Alex May from the Institute for Quantum Computing and Perimeter Institute for Theoretical Physics, present novel techniques for establishing lower bounds on the entanglement cost of such computations. Their work addresses a significant gap in the field, as previous methods struggled to provide meaningful limits for many simple unitaries. By analysing controllable correlation and entanglement, Cleve, May et al. demonstrate non-trivial lower bounds for Haar random two-qubit unitaries and, crucially, determine lower bounds for commonly studied gates like DCNOT and the XX interaction, for which no such limits were previously known. Notably, their approach yields a tight lower bound for the gate, fully characterising its entanglement cost and offering insights applicable to noisy quantum systems.

Quantifying entanglement costs for universal non-local quantum computation is a crucial step towards practical applications

Researchers have developed new techniques to quantify the entanglement cost associated with non-local quantum computation, a method of distributed quantum processing that sidesteps the need to physically move qubits between locations. Understanding this entanglement cost is crucial for advancements in diverse fields including complexity theory, cryptography, and even theoretical investigations into gravity.
Previous methods for determining these costs were limited to specific scenarios, leaving the entanglement requirements of many fundamental quantum operations poorly understood. This work introduces two novel lower bound techniques applicable to any unitary transformation, based on evaluating controllable correlation and controllable entanglement.

The study demonstrates that these techniques yield meaningful lower bounds for Haar random two-qubit unitaries, representing a significant step towards a more general understanding of entanglement requirements. Importantly, the researchers have successfully calculated lower bounds for several commonly used two-qubit quantum gates, including CNOT, DCNOT, √SWAP, and the XX interaction, none of which were previously known.

This achievement addresses a long-standing open problem in the field of quantum information theory, providing concrete values for the entanglement needed to implement these essential building blocks of quantum circuits. Notably, the new techniques provide a tight lower bound for the CNOT gate, fully resolving its entanglement cost and offering a benchmark for evaluating other quantum operations.

These lower bounds exhibit desirable “parallel repetition” properties, meaning they remain valid even when the computation is repeated multiple times, and crucially, they also hold in practical scenarios where noise is present. This robustness is essential for building fault-tolerant quantum computers and secure quantum communication networks. The findings pave the way for optimizing quantum protocols and designing more efficient quantum technologies.

Quantifying Controllable Correlation and Entanglement for Lower Bound Estimation requires careful consideration of system parameters

Researchers developed two new lower bound techniques to evaluate entanglement cost in non-local quantum computation (NLQC) applicable to any unitary. These techniques centre on quantifying controllable correlation and controllable entanglement within a given unitary transformation. The study begins by defining a non-local computation scenario where Alice and Bob share entanglement, perform local operations, exchange a single round of communication, and then perform further local operations to simulate a direct interaction between systems.

This contrasts with a local computation where systems are physically brought together for direct interaction. To establish lower bounds, the work leverages Haar random two-qubit unitaries as test cases, consistently yielding non-trivial results. Crucially, the methodology extends to commonly studied quantum gates including the CNOT, and the XX interaction, for which no prior lower bounds were known.

For the gate, one technique provides a tight lower bound, fully determining its entanglement cost. The researchers calculated a dimension lower bound of 0.0.15, an entropy lower bound for any pure state resource of 0.0.17, and an entanglement lower bound of 0.0.20. The study’s innovation lies in its ability to provide generic lower bounds applicable to any unitary, unlike previous techniques limited to specific NLQC instances.

Evaluation involved analysing the entanglement requirements for simulating unitary transformations using NLQC protocols. The resulting lower bounds exhibit parallel repetition properties, maintaining validity even in noisy computational environments, and are applicable to scenarios involving quantum position-verification, complexity theory, and cryptographic applications. This work establishes a foundation for quantifying entanglement costs and assessing the security and efficiency of NLQC protocols.

Entanglement cost quantification via controllable correlation and entanglement for two qubit unitaries is a challenging problem

Two new lower bound techniques for quantifying entanglement cost in non-local quantum computation have been developed, applicable to any unitary operation. These techniques, based on controllable correlation and controllable entanglement, yield non-trivial lower bounds for Haar random two qubit unitaries.

Consequently, lower bounds were established for commonly studied two qubit quantum gates, including CNOT, DCNOT, √SWAP, and the XX interaction, none of which previously possessed known lower bounds. For the CNOT gate, one of the techniques provides a tight lower bound, completely determining its entanglement cost.

The research demonstrates that the resulting lower bounds exhibit parallel repetition properties and remain valid even within noisy computational settings. Analysis of the lower bound from controllable correlation reveals its effectiveness in simple cases, providing a foundational understanding of its capabilities.

Further evaluation of the lower bound derived from controllable entanglement includes a dimension lower bound and an entropy lower bound applicable to any pure state resource. Specifically, the dimension lower bound technique assesses the minimal resources required for computation, while the entropy lower bound quantifies entanglement based on the purity of quantum states.

These analyses, combined with the entanglement lower bound itself, offer a comprehensive assessment of entanglement costs. Evaluation of these lower bounds in simple cases confirms their applicability and provides a benchmark for more complex scenarios. The work establishes a significant step towards understanding the fundamental limits of entanglement in non-local quantum computation and its implications for diverse applications including quantum cryptography and complexity theory.

Quantifying entanglement cost for universal quantum gates is a crucial step towards scalable quantum computation

Researchers have developed new techniques to determine the minimum entanglement required for non-local quantum computation, a crucial factor in assessing the complexity and feasibility of quantum processes. These methods establish lower bounds on entanglement cost for any unitary operation, addressing a long-standing challenge in the field.

Previously, such lower bounds were limited to specific cases, leaving the entanglement requirements of many fundamental quantum gates unknown. The newly presented techniques, based on quantifying controllable correlation and entanglement, successfully calculate non-trivial lower bounds for Haar random two-qubit unitaries and, importantly, for commonly used quantum gates including CNOT, Hadamard, and the XX interaction.

Notably, a tight lower bound was achieved for the CNOT gate, fully defining its entanglement cost. These bounds also exhibit properties suitable for noisy quantum systems, enhancing their practical relevance. The authors acknowledge that their analysis relies on certain assumptions regarding the controllability of entanglement and correlation within the quantum systems.

Furthermore, while the techniques provide lower bounds, determining the absolute minimum entanglement cost for all unitaries remains an open problem. Future research may focus on refining these techniques to approach tighter bounds and exploring their application to more complex quantum operations and architectures, potentially informing the development of more efficient quantum communication protocols and computational strategies.