Short answer: Not with the known proofs, and very likely not. All existing BQP–hardness proofs for the quantum linear systems problem (QLSP/HHL) fundamentally use negative (or complex) matrix elements to encode quantum phases. If you forbid those and insist that every entry of A and of b is in {0,1}, the standard reduction to a generic quantum computation breaks down. Moreover, under natural extra promises that keep the problem well-posed, the resulting linear-systems task lies in much weaker classical complexity classes (indeed in AM, and for large subclasses in BPP), so it would not be BQP-complete unless one believes BQP ⊆ AM.

Here is the landscape in a bit more detail.

• What the BQP-hardness proofs use. In Harrow–Hassidim–Lloyd (and subsequent refinements such as Childs–Kothari–Somma), BQP-hardness is obtained by embedding a unitary circuit U into a linear system via a matrix of the form A = I − e−1/T U. The phase structure of U is crucial: negative or complex entries appear and are what enables quantum interference in the reduction. If you require A to have only nonnegative entries (let alone all entries in {0,1}), this construction is impossible. The same is true of Janzing–Wocjan’s matrix powers reduction: at a key point they need a negative matrix element to simulate a controlled-Z.

• Why the 0/1 restriction points toward easier (classical) complexity. If you also impose standard well-conditioning/sparsity promises to make the instance meaningful, several large subclasses of 0/1 matrices admit efficient classical algorithms:

  • SDD/SDDM/Laplacian-type systems (which can be written with small integer entries and, after a trivial identity shift, have off-diagonals of one sign) can be solved in nearly-linear time classically (Spielman–Teng; Koutis–Miller–Peng). These cover many 0/1 constructions that one would naturally try.
  • When A can be written in a way that makes A−1 a convergent Neumann series with nonnegative terms (e.g., A = I − P with ρ(P) < 1, which happens for adjacency matrices of DAGs or suitably scaled stochastic matrices), then A−1 = ∑k≥0 Pk and each entry is a sum over nonnegative contributions (counts of walks). Approximating such sums admits Stockmeyer-type approximate-counting protocols and hence lies in AM. In this regime there is no destructive interference to exploit.

  Taken together, these observations strongly suggest that, under a “no negative/complex entries” restriction, the induced QLSP is not BQP-hard unless BQP ⊆ AM.

• What is actually known. The standard BQP-completeness statements for QLSP assume general sparse Hermitian A with poly-bounded condition number and do not restrict the sign/phase of its entries. Those proofs do not go through if all entries must be nonnegative (and certainly not if they must be in {0,1}). Conversely, there is no published proof that the 0/1-restricted QLSP is BQP-hard; on the contrary, as noted above, with additional natural structural promises it falls in AM or even BPP. If you relax the restriction just slightly to allow signs (entries in {−1,0,1} or, more generally, allowing complex phases through a block-encoding), BQP-hardness comes back via the usual constructions.

• About b. Requiring b to have 0/1 entries does not help restore hardness: after normalization, |b⟩ is just a uniform superposition over some subset, which is easy to prepare and does not introduce phases.

Conclusion. The BQP-completeness of HHL/QLSP relies on the ability to encode phases via negative or complex matrix elements. Under a strict {0,1} entry restriction for both A and b, the known BQP-hardness proofs break, and for broad natural subclasses the problem admits AM (and often BPP) algorithms. Thus, unless one believes BQP ⊆ AM, the 0/1-restricted version should not be BQP-complete. As far as is publicly known, proving BQP-hardness under the {0,1} restriction remains open and would require fundamentally new ideas that somehow reintroduce interference without using negative/complex entries.