Boosting quantum annealer performance via sample persistence


We propose a novel method for reducing the number of variables in quadratic unconstrained binary optimization problems, using a quantum annealer (or any sampler) to fix the value of a large portion of the variables to values that have a high probability of being optimal. The resulting problems are usually much easier for the quantum annealer to solve, due to their being smaller and consisting of disconnected components. This approach significantly increases the success rate and number of observations of the best known energy value in samples obtained from the quantum annealer, when compared with calling the quantum annealer without using it, even when using fewer annealing cycles. Use of the method results in a considerable improvement in success metrics even for problems with high-precision couplers and biases, which are more challenging for the quantum annealer to solve. The results are further enhanced by applying the method iteratively and combining it with classical pre-processing. We present results for both Chimera graph-structured problems and embedded problems from a real-world application.

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  1. 1.

    The chip at our disposal had 1100 active qubits, a working temperature of 26 ± 5 mK, and a minimum annealing time of 20 \(\upmu \)s.

  2. 2.

    A gauge, in this context, implies multiplying each spin operator by \(\pm 1\).

  3. 3.

    We used the function fix_variables in D-Wave Systems’ SAPI 2.3.1, which is the solver API used to access the quantum annealer [9].

  4. 4.

    The value zero was excluded for the couplers but not for the biases, and we use this convention throughout the paper.

  5. 5.

    The reduced problems often consist of multiple connected components (see Sect. 3.4). We took advantage of this fact when evaluating the energy values for the states in each sample.


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The authors would like to thank Dominic Marchand, Pooya Ronagh, and Brad Woods for their insightful comments, Marko Bucyk for editing the manuscript, and Alex Selby for the use of his implementation of the Hamze–de Freitas–Selby (HFS) algorithm, available for public use on GitHub [45]. This work was supported by 1QBit and Mitacs.

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Correspondence to Hamed Karimi.

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Conflict of interest

Hamed Karimi is an academic intern, and Gili Rosenberg is an employee at 1QBit. 1QBit is focused on solving real-world problems using quantum computers. D-Wave Systems is a minority investor in 1QBit.


Appendix 1: Dependence on thresholds

Table 6 Mean number of fixed variables for different fixing_threshold (column: ‘F’) and elite_threshold (row: ‘E’) using SPVAR on Chimera graph-structured problems (with 1100 variables)

To illustrate the dependence of the number of fixed variables and SPVAR’s fixing success rate on fixing_threshold and elite_threshold, we present results for two problem sets in Tables 6 and 7. The fixing success rate was defined as the percentage of problems for which the method fixed variables only to their optimal value. Detailed results for two choices of parameters, for the same two problem sets, are presented in Fig. 4. In that experiment, SPVAR was applied on a quantum annealer sample obtained from 2500 annealing cycles. We defined the quantum annealer’s residual as the energy difference between the quantum annealer’s lowest energy solution and the best known solution (found by a heuristic solver; see Sect. 3 for more details), and the method’s residual as the difference in energy between the best known solution before fixing variables and after fixing variables.

We see that the quantum annealer was able to find the best known solution for most of the low-precision problems (top row), but unable to find the best known solution for almost all of the higher-precision problems (bottom row). The method’s residual was almost always zero, indicating that the method almost never fixed variables incorrectly. In the few cases in which it did fix variables incorrectly, the effective problem (after fixing variables) still conserved the first or second excited states (the energy spacing was exactly two, due to the integer biases and couplers). We also note that for those problems, the method’s residual was always lower than the quantum annealer’s residual. The mean fraction of variables fixed was 58–70%, showing that SPVAR was able to fix most of the variables in these problems, and, as described in Sect. 3.2, the iterative method presented (ISPVAR) was able to fix even more variables.

Table 7 SPVAR fixing success rate for different fixing_threshold (column: ‘F’) and elite_threshold (row: ‘E’) on Chimera graph-structured problems
Fig. 4

(Colour online) Results for SPVAR on Chimera graph-structured problems. a and b Present results for problems in \(U_2\) (\(J,h \in \{-2,-1,0,1,2\}\), excluding 0 for J), and c and d present results for problems in \(U_{10}\). The parameters are indicated in the titles above each subfigure. The top panel of each subfigure shows the difference in energy between the quantum annealer’s best solution and the best known solution, for each problem, obtained from 2500 annealing cycles using five random gauges. The middle panel shows the difference in energy between the best known solution before fixing variables and the best known solution after fixing variables (using the same 2500 cycles)—it should be zero if all variables fixed by SPVAR were fixed correctly. The bottom panel shows the number of fixed variables for each problem. A horizontal dashed line indicates the mean number of fixed variables. In all panels, the points that correspond to problems for which SPVAR fixed variables incorrectly are shown in red, and the rest are shown in blue

Appendix 2: Parameter description

In Table 8, we list the parameters used in the text and give a short description of each.

Table 8 Parameters of SPVAR and ISPVAR

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Karimi, H., Rosenberg, G. Boosting quantum annealer performance via sample persistence. Quantum Inf Process 16, 166 (2017).

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  • Adiabatic quantum computation
  • Quantum annealing
  • Variable reduction
  • Sample persistency
  • Binary optimization