  We define a map from an arbitrary quantum circuit to a local Hamiltonian
whose ground state encodes the quantum computation. All previous maps relied on
the Feynman-Kitaev construction, which introduces an ancillary `clock register'
to track the computational steps. Our construction, on the other hand, relies
on injective tensor networks with associated parent Hamiltonians, avoiding the
introduction of a clock register. This comes at the cost of the ground state
containing only a noisy version of the quantum computation, with independent
stochastic noise. We can remedy this - making our construction robust - by
using quantum fault tolerance. In addition to the stochastic noise, we show
that any state with energy density exponentially small in the circuit depth
encodes a noisy version of the quantum computation with adversarial noise. We
also show that any `combinatorial state' with energy density polynomially small
in depth encodes the quantum computation with adversarial noise. This serves as
evidence that any state with energy density polynomially small in depth has a
similar property. As applications, we give a new proof of the QMA-completeness
of the local Hamiltonian problem (with logarithmic locality) and show that
contracting injective tensor networks to additive error is BQP-hard. We also
discuss the implication of our construction to the quantum PCP conjecture,
combining with an observation that QMA verification can be done in logarithmic
depth.
