How does relative decoding broaden quantum Gibbs state preparation guarantees?

Ordinary decoding caps the certified inverse temperature

Preparing the Gibbs state ρβ(H) = e−βH / Tr e−βH of a many-body Hamiltonian is a central task in quantum simulation. Two routes dominate: quantum Gibbs samplers, whose fixed point is ρβ(H), and coherent constructions that prepare the thermofield double (TFD), the canonical purification of ρβ(H). Their cost is governed by a mixing time or a spectral gap, and provable bounds exist only for restricted families, such as local Hamiltonians at high temperature or finite-range spin chains at every fixed temperature.

Decoded quantum interferometry (DQI) and its Hamiltonian version HDQI take a different route. They prepare states whose amplitudes are a polynomial Q_D of the Hamiltonian by combining Fourier interference with coherent decoding of a classical code attached to the Hamiltonian; choosing Q_D ≈ e^{−βx/2} gives an approximate TFD. Writing the Hamiltonian as a sum of m Pauli terms, the subsets of terms whose product is proportional to the identity form a binary linear code K_B = ker B, with the Pauli-label matrix B as parity-check matrix. Expanding Q_D(H_B) gives a sum over subsets z ∈ F_2^m of at most D terms; two subsets produce the same Pauli label Bz exactly when they differ by a relation, and recovering z from Bz is syndrome decoding.

That decoding is unambiguous only if 2D < d_ord, where d_ord is the fewest terms in any target relation. This is the bottleneck of DQI and HDQI: the certified inverse temperature is set by the polynomial approximation and the decoding radius, and the standard exact-decoding guarantee for a degree-D filter is 2D < d_ord. The paper's observation is that z itself is never needed.

Relative distance replaces ordinary distance in the decoding window

The transduction circuit only has to erase the source label Az, which is already blind to source relations. When K_A ⊆ K_B, and the anticommutation patterns and relation phases of the two Hamiltonians agree, Bz determines Az for all |z| ≤ D as soon as 2D < d_rel, where d_rel := min_{c ∈ K_B \ K_A} |c| is the distance of the quotient code K_B/K_A.

Always d_rel ≥ d_ord, with strict inequality exactly when every minimum-weight target relation is already a source relation. Absorbing the short target relations can therefore enlarge the decoding window and, with an efficient relative decoder, certify lower temperatures. DQI and HDQI are the relation-free case K_A = 0.

The distinction matters because d_ord and d_rel answer different questions. d_ord bounds the uniform exact decoding radius of DQI and HDQI, where the decoder must resolve every target relation. d_rel bounds only the relations absent from the source, so it measures the residual ambiguity after source relations have been identified. The paper states that the largest achievable d_rel for a fixed source or a fixed target has an exact coset description, and that a minimum-weight ordinary decoder gives a valid relative decoder throughout the relative window.

How the transduction circuit prepares the target TFD

Starting from the canonical TFD of H_A, the construction applies a Bell transform, a reversible label map, and a coherent decoder for the quotient code K_B/K_A to prepare an approximate TFD of H_B, and hence its Gibbs state. The decoder only resolves target relations absent from the source, which is why d_rel rather than d_ord controls the reachable inverse temperature.

Theorem 1.1 makes the guarantee explicit. Let H_A and H_B have the same anticommutation pattern, K_A ⊆ K_B, and matching relation phases, and let a coherent relative decoder for K_B/K_A be efficient up to degree D_dec < d_rel/2. Given the canonical TFD of H_A at inverse temperature β to trace distance δ_src, an efficient circuit prepares the TFD of H_B, and hence its Gibbs state, to trace distance δ_src + 2η + δ_circ whenever D_dec ≥ βΛ + O(√(βΛ ln(1/η)) + ln(1/η)), with Λ ≥ ‖H_A‖.

In particular, D_dec = Ω(m) and Λ = O(m) certify every inverse temperature below a fixed constant. For an extensive spectral bound, efficient relative decoding at linear radius certifies a constant-inverse-temperature preparation window. The error is stated as a trace-distance bound, so the guarantee is about the prepared state's distance from the target TFD, not about a mixing time of any dynamics.

A concrete target family with d_ord = 3 and d_rel = Θ(m)

Theorem 1.2 gives the main example. Fix N and real coefficients λ_i^μ with 0 < |λ_i^μ| ≤ J_0. The source is the nearest-neighbor chain on N two-qubit cells (q_i, c_i), H_A = Σ_{i=1}^{N} (λ_i^x X_{q_i} + λ_i^z Z_{q_i} + λ_i^c Z_{c_i} + λ_i^{cq} Z_{c_i} Z_{q_i}) + Σ_{i=1}^{N−1} λ_i^{zz} Z_{q_i} Z_{q_{i+1}}, whose canonical TFD is efficiently preparable at every fixed temperature.

Let F ∈ F_2^{(N−k)×N} be a parity-check matrix with columns f_1, …, f_N, distance d_F ≥ δN, and an efficient syndrome decoder that recovers every error of weight at most ⌊τN⌋, where 0 < 2τ < δ are constants. Replace the N central qubits by N−k syndrome qubits R_1, …, R_{N−k} and each Z_{c_i} by the parity check Z_R(f_i) := Π_{a: (f_i)_a = 1} Z_{R_a}, so that H_B = Σ_{i=1}^{N} (λ_i^x X_{q_i} + λ_i^z Z_{q_i} + λ_i^c Z_R(f_i) + λ_i^{cq} Z_R(f_i) Z_{q_i}) + Σ_{i=1}^{N−1} λ_i^{zz} Z_{q_i} Z_{q_{i+1}}.

Then H_A and H_B are compatible, with m = 5N − 1 terms, d_ord = 3, d_rel = d_F = Θ(m), and D_dec = ⌊τN⌋, and relative decoding is exactly syndrome decoding of F. The TFD and Gibbs state of H_B are prepared in polynomial time for every fixed β < β_0 = Θ(τ/J_0). If F has bounded row and column weights, H_B has bounded-weight terms and bounded qubit degree; if the supports of the columns of F have unbounded diameter, its interactions are geometrically nonlocal.

Physically, H_A is a mixed-field Ising chain on the spins q_i, with transverse field λ_i^x, longitudinal field λ_i^z and coupling λ_i^{zz}, in which each longitudinal field is shifted by λ_i^{cq} times a classical spin Z_{c_i}. In H_B that classical spin is replaced by a parity Z_R(f_i) of the syndrome spins: the quantum sector is unchanged and remains one-dimensional, while the classical sector becomes a parity-check spin system whose checks drive the longitudinal fields of the chain.

Why the source must already contain the short relations

In the chain example, every cell carries the weight-three relation Z_{q_i} · Z_{c_i} · Z_{c_i} Z_{q_i} = I and every bond the relation Z_{q_i} · Z_{q_{i+1}} · Z_{q_i} Z_{q_{i+1}} = I, so d_ord = 3 and the ordinary exact-decoding guarantee for HDQI on H_B reaches only degree one. The chain H_A already contains all of these short relations, and its TFD is efficiently preparable. The quotient code K_B/K_A then consists exactly of the codewords of F, so d_rel = d_F, and the coherent relative decoder is the classical decoder for F run reversibly.

This is the structural reason the construction works. To increase the relative distance, the source should absorb the low-weight target relations, and the resulting degree window can be used when source preparation and relative decoding remain efficient. The paper also notes that finite d_rel rules out termwise unitary conjugation: if a unitary satisfies the termwise relation for every i, then K_A = K_B and d_A→B = +∞, so d_A→B < ∞ implies K_A ⊊ K_B and no termwise unitary conjugation exists. Every finite residual distance belongs to {1, …, m}.

For a fixed source, linear relative distance is possible precisely when K_Σ/K_A contains a coset at linear distance from K_A. For a fixed target and a proper source kernel, it is possible when the target relations below the desired linear threshold span a proper subspace of K_B. Source design therefore balances absorbing short target relations against retaining an efficiently preparable source TFD.

What the construction does not yet settle

The guarantee is conditional on a source TFD that is already efficiently preparable and on a coherent relative decoder that is efficient up to the required degree. The paper's main example uses a nearest-neighbor chain whose canonical TFD is known to be preparable at every finite temperature, and a classical parity-check matrix with an efficient syndrome decoder; both ingredients are inputs to the construction rather than consequences of it.

The certified temperatures are bounded: in the chain example the preparation window is every fixed β < β_0 = Θ(τ/J_0), where τ comes from the decoder's error-correction radius and J_0 bounds the coefficients. The paper also states that further constructions can use sources with different relation structures and efficient relative decoders to reach other temperature and accuracy regimes, and that the homomorphic formulation supports fermionic and qudit systems as well as bosonic models with controlled finite-dimensional truncations, but thermal applications in those settings call for explicit source TFDs and relative decoders.

Two goals are named as future work: establishing classical sampling hardness for target Gibbs states at the certified temperatures, and comparing transduction with direct target-preparation methods. A family with polynomial-time transduction and exponentially slow mixing for a specified target sampler at the same temperature would demonstrate a separation between preparation and equilibration by that dynamics.

The priority is the relative window, not the target distance

The decision-relevant comparison in this paper is between d_ord and d_rel, not between different target families. A target with d_ord = 3 would, under the ordinary exact-decoding guarantee, support only degree-one decoding; the same target becomes preparable at a constant inverse temperature once the source absorbs its short relations and the decoder only has to resolve K_B/K_A, giving d_rel = Θ(m) and D_dec = ⌊τN⌋.

So the practical question a reader should ask of any candidate source–target pair is not how large the target's ordinary distance is, but how much of the target's relation code the source already contains. The paper's structural results make that question checkable: for a fixed anticommutation pattern Σ, exactly the chains K_A ⊆ K_B ⊆ ker Σ are realizable, and the largest achievable d_rel for a fixed source or target has an exact coset description. The construction pays off when the source's TFD is efficiently preparable and the residual quotient code admits an efficient decoder at linear radius; when either condition fails, the relative window does not translate into a temperature guarantee.

Sources used in this answer