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DTSTART:20261101T010000
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DESCRIPTION:The recent resolution of the NLTS Conjecture [ABN22] establishes a prerequisite to the Quantum PCP (QPCP) Conjecture through a novel use of newly-constructed QLDPC codes [LZ22]. Even with NLTS now solved\, there remain many independent and unresolved prerequisites to the QPCP Conjecture\, such as the NLSS Conjecture of [GL22]. In this talk we focus on a specific and natural prerequisite to both NLSS and the QPCP Conjecture\, namely\, the existence of local Hamiltonians whose low-energy states all require ω(log n) T gates to prepare. In fact\, we will show a stronger result which is not necessarily implied by either conjecture: we construct local Hamiltonians whose low-energy states require Ω(n) T gates. We further show that our procedure can be applied to the NLTS Hamiltonians of [ABN22] to yield local Hamiltonians whose low-energy states require both Ω(log n)-depth and Ω(n) T gates to prepare. This result represents a significant improvement over [CCNN23] where we used a different technique to give an energy bound which only distinguishes between stabilizer states and states which require a non-zero number of T gates.
X-ALT-DESC;FMTTYPE=text/html:<span dir="ltr">The recent resolution of the NLTS Conjecture [ABN22] establishes a prerequisite to the Quantum PCP (QPCP) Conjecture through a novel use of newly-constructed QLDPC codes [LZ22]. Even with NLTS now solved, there remain many independent and unresolved prerequisites to the QPCP Conjecture, such as the NLSS Conjecture of [GL22]. In this talk we focus on a specific and natural prerequisite to both NLSS and the QPCP Conjecture, namely, the existence of local Hamiltonians whose low-energy states all require ω(log n) T gates to prepare. In fact, we will show a stronger result which is not necessarily implied by either conjecture: we construct local Hamiltonians whose low-energy states require Ω(n) T gates. We further show that our procedure can be applied to the NLTS Hamiltonians of [ABN22] to yield local Hamiltonians whose low-energy states require both Ω(log n)-depth and Ω(n) T gates to prepare. This result represents a significant improvement over [CCNN23] where we used a different technique to give an energy bound which only distinguishes between stabilizer states and states which require a non-zero number of T gates.</span>
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SUMMARY:Hamiltonians whose low-energy states require Ω(n) T gates
DTSTART;TZID=America/New_York:20240305T150000
DTEND;TZID=America/New_York:20240305T160000
DTSTAMP:20260424T205213Z
TRANSP:OPAQUE
STATUS:CONFIRMED
SEQUENCE:0
LOCATION:Zoom + QNC 1201
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