  We present an algorithm that uses block encoding on a quantum computer to
exactly construct a Krylov space, which can be used as the basis for the
Lanczos method to estimate extremal eigenvalues of Hamiltonians. While the
classical Lanczos method has exponential cost in the system size to represent
the Krylov states for quantum systems, our efficient quantum algorithm achieves
this in polynomial time and memory. The construction presented is exact in the
sense that the resulting Krylov space is identical to that of the Lanczos
method, so the only approximation with respect to the exact method is due to
finite sample noise. This is possible because, unlike previous quantum Krylov
methods, our algorithm does not require simulating real or imaginary time
evolution. We provide an explicit error bound for the resulting ground state
energy estimate in the presence of noise. For our method to be successful
efficiently, the only requirement on the input problem is that the overlap of
the initial state with the true ground state must be $\Omega(1/\text{poly}(n))$
for $n$ qubits.
