Fun with QMA and non-unitary gates
Dan Browne joint work with: Naïri Usher QMA
❖ Kitaev (1999) - QMA: The quantum analogue of NP QMA for beginners
talk contains Interstellar spoilers - sorry! NP
Arthur Merlin
Poly-time classical Unbounded computer computational power NP
Arthur Merlin
Decision problem:
“Is there an answer to the problem of gravity?”
Arthur can verify proof “Yes” validity of proof “And here’s the proof” in poly-time NP
Arthur Merlin
Decision problem:
“Is there an answer to the problem of gravity?” “No” “And there is no Arthur can verify fake proof I could validity of proof send to you to trick in poly-time you into thinking the answer is yes” NP
Arthur Merlin
Poly-time classical Unbounded computer computational power
Decision problem proof “Yes = 1” Arthur can verify “And here’s the proof” validity of proof in poly-time “No = 0” “And there is no fake proof …” NP MA
Arthur Merlin
Poly-time classical Unbounded computer computational power
Decision problem proof “Yes = 1” Arthur can verify “And here’s the proof” validity of proof with “high in poly-time enough” “No = 0” probability “And there is no fake proof …” “Is there an answer to the problem of gravity?”
We need quantum data! SPOILER: “I lied. It is not in NP.”
NP GRAV QMA
❖ Kitaev (1999) - QMA: The quantum analogue of NP QMA
Arthur Merlin
Poly-time quantum Unbounded computer computational power
Decision problem quantum proof “Yes = 1” Arthur can verify “And here’s the proof” | i validity of proof with “high in poly-time enough” “No = 0” probability “And there is no fake proof …” Local Hamiltonian Problem
PP Kitaev (1999):
LH: k-local Hamiltonian QMA LH Problem is QMA- complete GRAV
NP 3-SAT Local Hamiltonian Problem
Given: A k-local n-qubit Hamiltonian 0 Hj 1 H = H j Hj : k-local j X with a promise that the ground state energy of H is:
E0 >b E = b or “promise gap” >1/poly n E = a 0 Problem: Determine whether E0 > b or E0 < a? LH is QMA-complete LH is in QMA ground state of H | i Can efficiently estimate the energy on a quantum computer. LH is hard for QMA Encode proof verification for any QMA problem into a ground state of a LH quantum proof | i poly-time verification ground state Yes instances: Low energy No instances: Higher energy Encoding a computation in a ground state Feynman’s History state idea History State = 0 | i | ini| i Unitary Gates +U 1 1| ini| i +U U 2 2 1| ini| i +U U U 3 3 2 1| ini| i + ··· Encoding a computation in a ground state Propagator Hamiltonian: 1 Hj = (Uj j j 1 +h.c.)+ Hprop = Hj 2 ⌦ | ih | j 1 X ( j j + j +1 j +1) 2 ⌦ | ih | | ih | 0 | ini| i +U 1 Ground space is 1| ini| i space of history states: +U2U1 in 2 | i| i +U U U 3 3 2 1| ini| i + ··· Encoding a computation in a ground state This works, because the ground space of 1 H = (U j j 1 +h.c.)+ j 2 j ⌦ | ih | 1 ( j j + j +1 j +1) 2 ⌦ | ih | | ih | is: j 1 + U j | i| i j| i| i Encoding a computation in a ground state This works, because the ground space of 1 H = (U j j 1 +h.c.)+ j 2 j ⌦ | ih | 1 ( j j + j +1 j +1) 2 ⌦ | ih | | ih | is: H ( j 1 + U j ) j | i| i j| i| i =(1/2)( U j + U j )=0 j| i| i j| i| i QMA Arthur Merlin Poly-time quantum Unbounded computer computational power Decision problem quantum proof “Yes = 1” Arthur can verify “And here’s the proof” | i validity of proof with “high in poly-time enough” “No = 0” probability “And there is no fake proof …” Encoding a computation in a ground state The verification circuit we wish to encode: Z meas Proof: | i Unitary Gates 0 Ancillas: |0i |0i | i L = No. of unitary gates in circuit Encoding a computation in a ground state Maps into the ground space of H = Hin + Hprop + Hout Z meas = 1 Proof: | i Unitary Gates 0 Ancillas: |0i |0i | i H = 1 1 0 0 in | ijh | ⌦ | iclockh | j ancilla 2X H = 0 0 L L out | iouth | ⌦ | iclockh | Encoding a computation in a ground state Yes instance: H = Hin + Hprop + Hout No frustration: Each term of Hamiltonian can reach its ground space: Z meas Proof: | i Ground state: Unitary Gates History state 0 Ancillas: |0i of this circuit |0i | i Encoding a computation in a ground state No instance: H = Hin + Hprop + Hout Frustration: Hout wants output qubit to be in state 1, but no proof state exists to allow this. Z meas Proof: | i Unitary Gates Ground state energy 0 must be increased. Ancillas: |0i |0i | i Promise gap scaling LH Problem: Determine whether E0 > b or E0 < a? E0 >b E = b or “promise gap” >1/poly n E = a 0 We need the “promise gap” between yes and no to remain inverse polynomial. Encoding a computation in a ground state Kitaev: Energy of no frustrated ground state instance scales with E 2 sin2(✓/2) 0 1 scales with 1/L 2nd lowest eigenvalue of Hprop Kitaev’s Hprop: ⇡ Lowest eigenvalue: 1 cos(⇡/L + 1) ⇡ 2(L + 1)2 LH is QMA-complete Non-unitary ground state computation? Our work: (Usher/Browne, unfinished 2015) Non-unitary ground state computation? Our work: (Usher/Browne, unfinished 2015) What if? Unitary Gates Ground State + non-Unitary projectors Computation Add projectors to the Verifier circuits circuits Non-unitary ground state computation? Renormalised Projectors / Post-selection E.g. quantum lottery ticket 1 win + lose 10000000000| i | i win win | ih | 1 win 10000000000| i Renormalise win | i Non-unitary ground state computation? Why? • Curiosity! - Feynman’s construction pre-dates all quantum computing theory. • Add projectors to unitary circuits and you get: Fault-tolerant Quantum Computation Measurement-based Quantum Computation postIQP = postBQP = PP Non-unitary ground state computation? Why it will never work! PP postIQP = postBQP = PP QMA Aaronson: LH If deterministic projectors (post-selection) are added to unitary gates, we can efficiently solve PP-hard problems. NP Non-unitary ground state computation? Why it might just work….. postIQP = postBQP = PP + + | i | ih | 1 + H | i p2 | i One-bit teleportation circuit (Zhou / Leung / Chuang 2000). Non-unitary ground state computation? Why it might just work….. postIQP = postBQP = PP + + | i | ih | 1 + H | i p2 | i One-bit Zero-bit teleportation circuit. NB Norm of output is independent of input. Encoding a computation in a ground state Why it also might just work….. In Kitaev construction, Hout projects the input to the correct proof state. H = Hin + Hprop + Hout Z meas Proof: Accepting | i circuit Unitary Gates projects 0 Ancillas: |0i input into |0i proof state | i The projector gadget Recall Kitaev and Feynman’s unitary gadget: 1 H = (U j j 1 +h.c.)+ j 2 j ⌦ | ih | 1 ( j j + j +1 j +1) 2 ⌦ | ih | | ih | that had ground space j 1 + U j | i| i j| i| i A projector gadget? For projector P, want a gadget Hp =? that has ground space P j where j 1 + | i| i | i| i = P j k | i| ik To avoid PP-hardness, we assume β will be equal for all states |Ψ⟩. The projector gadget 2 1 1 1 H = P j 1 j j j 1 + j 1 j 1 + j j + P ? j j j 1+ 2 ⌦ | ih | | ih | 2 | ih | | ih | ⌦ | ih | ✓ ◆ ✓ ◆ has the ground space we want! P j j 1 + | i| i | i| i NB In the limit β →1, P→�, we recover Kitaev/Feynman gadget. The projector gadget 2 1 1 1 H = P j 1 j j j 1 + j 1 j 1 + j j + P ? j j j 1+ 2 ⌦ | ih | | ih | 2 | ih | | ih | ⌦ | ih | ✓ ◆ ✓ ◆ P j H j 1 + | i| i j | i| i ✓ ◆ P P j + j =0 / | i| i | i| i The projector gadget 2 1 1 1 H = P j 1 j j j 1 + j 1 j 1 + j j + P ? j j j 1+ 2 ⌦ | ih | | ih | 2 | ih | | ih | ⌦ | ih | ✓ ◆ ✓ ◆ has the ground space we want! P j j 1 + | i| i | i| i NB In the limit β →1, P→�, we recover Kitaev/Feynman gadget. Encoding a computation in a ground state With this gadget, we can construct Hamiltonians H = Hin + Hprop + Hout such that “Yes” instances have low energy… Z meas Proof: | i Ground state: Unitary and History state 0 Projector Gates Ancillas: |0i of this circuit |0i | i Encoding a computation in a ground state And No instances have frustration H = Hin + Hprop + Hout The “promise gap” satisfies Kitaev’s formula E 2 sin2(✓/2) 0 1 Need to check this! should scale as 1 / poly(n) 2nd lowest eigenvalue of Hprop Characterising the promise gap December 2014: We had a beautiful and elegant analytic bound on λ1 for many circuits. January 2015: We found one of these in the proof… Characterising the promise gap Instead - some quick and simple proof-of-principle numerics. + + | i | ih | + + + | i | ih | + + + | i | ih | + | i Equivalent to H H H | i Characterising the promise gap 1/λ1 1200 1000 1 EXPONENTIAL!! :-( 800 1.14(n+1) 600 Fit: 2 Inverse of λ 400 200 3 4 5 6 7 8 n: Number of 1-bit teleportations in circuit Characterising the promise gap Extra ancillas giving low energy excited states. :-( + + | i | ih | + + + | i | ih | + + + | i | ih | + | i Characterising the promise gap A second try + + + + | i | ih | | ih | + + + + + | i | ih | | ih | Equivalent to H H H H | i Characterising the promise gap 600 1/λ1 500 QUADRATIC!! :-) 1 400 Fit: 1.4+0.088x +5.67x2 300 200 Inverse of λ 100 2 4 6 8 10 n: Number of 1-bit teleportations in circuit Conclusion It works! Sometimes….. (probably) Where do we go next? Where do we go next? ❖ Numerical evidence for 1/poly promise gap scaling: ❖ But when does exponential scaling occur? ❖ Analytic bounds? ❖ Aim: A QMA construction from a non-universal gate set with projectors (e.g. IQP circuit). Where do we go next? ❖ Complexity ❖ When does BQP + projectors = BQP ❖ Is constant probability on input states sufficient? ❖ Can we characterise exponentially closing gap with other complexity classes? Where do we go next? ❖ Post-selection gives IQP / MBQC circuits trivial time complexity. QMA with constant clock-steps? + + | i | ih | + + + Need many clocks! | i | ih | + + + | i | ih | + | i Where do we go next? ❖ Incorporate fault tolerance? ❖ Error detection gadgets to project onto error free states? ❖ Robustness of Hamiltonian to perturbations? ❖ Norm of history state vectors problematic? + + + | i | ih | Syndrome Where do we go next? ❖ Other applications for the projector gadget? ❖ Adiabatic Quantum Computation? ❖ Relationship with Bacon and Flammia’s adiabatic cluster state model? Thank you!