Researchers from University College London, Stanford University, and Instituto de Ciencias Matemáticas Madrid have explored the link between position-based quantum cryptography (PBQC) and holography. Using holographic quantum error correcting codes as models, they discovered that if PBQC is secure against attacks with small entanglement, there are new fundamental lower bounds for resources required for one Hamiltonian to simulate another. This could potentially lead to new lower bounds on achievable simulations. The study also found that to obtain a model with a consistent causal structure, the time component of the metric needs to be fixed manually.
Introduction to Position-Based Quantum Cryptography and Holography
The research conducted by explores the connection between position-based quantum cryptography (PBQC) and holography.
The researchers used holographic quantum error correcting codes as toy models to investigate this link. They found that if the temporal scaling of the AdS metric is inserted into the toy model via the bulk Hamiltonian interaction strength, a toy model with a consistent causality structure is recovered. This discovery implies that if PBQC is secure against attacks with small entanglement, there are new fundamental lower bounds for resources required for one Hamiltonian to simulate another.
Causal Structure of Toy Models of AdS/CFT
The researchers used toy models of AdS/CFT correspondence to encode the spatial component of the AdS metric. These models capture various features of AdS/CFT while being exactly solvable. The bulk theory acts on a free index of each tensor, while the boundary theory acts on the dangling indices at the truncation of the tessellation. The researchers argue that to obtain a toy model with consistent causal structure, the time component of the metric needs to be fixed by hand. By selecting a bulk Hamiltonian that incorporates this time dilation, the researchers demonstrated that the resulting tensor network model of holography exhibits the anticipated causal structure.
Secure Finite Dimensional PBQC and Upper Bound on Simulation Techniques
The researchers studied position-based cryptography protocols in tensor network toy models of AdS/CFT that have the improved causality structure. While the construction does not provide a holography-inspired PBQC protocol that beats the best known quantum information protocols, it does allow the researchers to relate two seemingly distinct fields of study in quantum information: PBQC and Hamiltonian simulation. The researchers showed that if PBQC is secure against attacks with linear entanglement, this implies fundamental limits on Hamiltonian simulation techniques that recreate the time dynamics of target Hamiltonians.
The Connection Between Hamiltonian Simulation and PBQC
The researchers found an interesting implication between two topics in quantum information. If position-based cryptography is secure against attacks with small entanglement, then there are new fundamental lower bounds for resources required for one Hamiltonian to simulate another. This discovery could potentially lead to new lower bounds on achievable simulations.
Discussion on Toy Models of AdS/CFT
The researchers used toy models of AdS/CFT to investigate the link between PBQC and holography. These models provide an encoding map from bulk to boundary and vice versa, realizing various aspects of the correspondence explicitly. However, the proposed dictionaries lead to superluminal signalling on the boundary. The researchers argue that to obtain a toy model with consistent causal structure, the time component of the metric needs to be fixed by hand.
Motivation of Matching Lower Bound for Boundary Entanglement
The researchers were motivated to construct tensor network toy models of AdS/CFT with the correct causality structure to investigate PBQC within these models. They argued that if arbitrary computations can occur in the bulk of AdS, then the bulk-boundary correspondence indicates that PBQC can be broken using only linear entanglement. This is exponentially better than the best-known quantum information protocols.
Explicit Non-Perturbative and Perturbative Construction of an Attack on PBQC
The researchers provided explicit non-perturbative and perturbative constructions of an attack on PBQC. They showed that if PBQC is secure against attacks with linear entanglement, this implies fundamental limits on Hamiltonian simulation techniques that recreate the time dynamics of target Hamiltonians. This discovery could potentially lead to new lower bounds on achievable simulations.
“Security of position-based quantum cryptography limits Hamiltonian simulation via holography” was published on January 17, 2024, by authors Harriet Apel, Toby S. Cubitt, Patrick Hayden, Tamara Kohler, and David Pérez-Garcı́a. The article was sourced from arXiv (Cornell University).
