Scalable Emulation of Stoquastic Hamiltonians with Room Temperature p-bits
The growing field of quantum computing is based on the concept of a q-bit which is a delicate superposition of 0 and 1, requiring cryogenic temperatures for its physical realization along with challenging coherent coupling techniques for entangling them. By contrast, a probabilistic bit or a p-bit is a robust classical entity that fluctuates between 0 and 1, and can be implemented at room temperature using present-day technology. Just as three-terminal transistors provide a building block for large functional circuits, a three terminal realization of the p-bit can provide a building block for p-circuits that has been shown to perform a variety of useful functions. In this paper, we introduce a new application of p-circuits, namely to perform Path Integral Monte Carlo (PIMC) simulations of quantum systems based on the well-known Suzuki-Trotter decomposition that maps a -dimensional quantum many body Hamiltonian to a +1-dimensional classical Hamiltonian. This approach allowing a quantum system to be emulated by a number of classical replicas is commonly used in software or high-level hardware simulations. What we show here is that it should be possible to build low-level hardware accelerators that use replicated p-bits to emulate a given q-bit network. Using full device-level SPICE simulations we demonstrate that the correct quantum correlations can be obtained using a classical p-circuit built with existing technology and operating at room temperature. Another advantage of p-bit networks is that they can be interconnected using conventional electronic devices such as GPUs or FPGAs. This could allow complex connectivity beyond nearest neighbor coupling as well as the implementation of arbitrary -body interactions with .
The basic building block of conventional digital electronics is the CMOS (Complementary Metal Oxide Semiconductor) transistor that is used to represent deterministic bits, that are either 0 or 1. Quantum computing, on the other hand, is based on q-bits that are coherent, delicate superpositions of 0 and 1. It is possible to define an entity intermediate between bits and q-bits that are classical but probabilistic, which we call “p-bits” Camsari et al. (2017a). It has been argued that just as three-terminal transistors provide a building block for large functional circuits, a three terminal realization of the p-bit can provide a building block for p-circuits Behin-Aein et al. (2016) reminiscent of the probabilistic computer described by Feynman in the same paper that helped launch the field of quantum computing Feynman (1982).
Such p-circuits can perform useful functions broadly relevant in the context of quantum computing and machine learning Camsari et al. (2018). For example, p-circuits can be used to perform classical annealing in hardware Sutton et al. (2017), perform integer factorization by operating multipliers in an invertible mode Camsari et al. (2017a); Pervaiz et al. (2017a), just like quantum annealers that have been used for similar applications Martoňák et al. (2004); Peng et al. (2008). In the machine learning context, p-bits can function as hardware accelerators for binary stochastic neurons Ackley et al. (1985) that can be used to become efficient inference engines Faria et al. (2018); Zand et al. (2018), or they can be used in an efficient calculation of correlations to accelerate learning algorithms, an application area also discussed in the context of quantum computing Bian et al. (2010); Adachi and Henderson (2015); Liu et al. (2018); Amin et al. (2018).
In this paper, we introduce a new application of p-circuits, namely to perform Path Integral Monte Carlo (PIMC) simulations of quantum systems based on the well-known Suzuki-Trotter decomposition that maps a -dimensional quantum many body Hamiltonian to a +1-classical Hamiltonian. This allows a quantum system to be emulated by a number of classical replicas that are interacting with each other Suzuki (1976) (FIG. 1) and this approach is commonly used in software or high-level hardware simulations Santoro et al. (2002); Heim et al. (2015); Denchev et al. (2016); Baldassi and Zecchina (2018); Okuyama et al. (2017). By contrast, we show that p-bits realized as low-level hardware building blocks to build p-computers can significantly speed up the simulation.
We show that for a class of quantum Hamiltonians generally referred to as stoquastic Hamiltonians Albash and Lidar (2018a) which avoid the sign problem Troyer and Wiese (2005) and are amenable to efficient PIMC simulation, it should be possible to build hardware accelerators using replicated p-bits to emulate the thermodynamics of q-bit networks. The number of p-bits required to emulate a given q-bit network is typically a factor of 25-100 larger, but this is offset by the relative ease of implementation. Three-terminal p-bits are straightforwardly implemented at room temperature with Magnetoresistive Random Access Memory (MRAM) technology which is currently in production with nearly Gb’s of memory elements. Non-magnetic and completely digital implementations of p-bits are also possible Pervaiz et al. (2017a, b) though they would require much larger energy and area Zand et al. (2019).
A p-bit based quantum emulator could overcome fundamental difficulties associated with the low temperature operation of quantum annealers Albash et al. (2017). It has recently been suggested Albash and Lidar (2018b) that among quantum annealing options, simulated quantum annealing (SQA) exhibits the best scaling properties, performing even better than experimental quantum annealers in some cases. As such, accelerating the software-based SQA with very low-level hardware is desirable.
Another advantage of p-bit networks is that unlike q-bit networks they can be interconnected using conventional electronic devices such as GPUs or FPGAs. This could allow all-to-all connectivity beyond nearest neighbor coupling without requiring any special encoding Lechner et al. (2015); De las Cuevas and Cubitt (2016). Moreover, it should allow the implementation of arbitrary -body interactions that are usually avoided by introducing ancillary bits to map them into 2-body interactions Biamonte (2008); Jiang et al. (2018).
Organization of the paper
We start in Section II, with a description of the mapping from the q-bit network to the p-bit network, along with the behavioral equations describing the dynamics of the latter. These behavioral equations for p-circuits are similar to those used for stochastic neural networks and are often implemented in software for machine learning applications. However, a hardware implementation can provide a significant speed-up especially because it can allow parallel asynchronous operation under the right conditions.
Next in Section III we consider one popular example of a stoquastic Hamiltonian, namely the Transverse Ising Hamiltonian Kadowaki and Nishimori (1998); Pfeuty (1970), commonly employed by quantum annealers Johnson et al. (2011). We compare the exact quantum results for the averages and correlations with the results obtained from the p-bit network demonstrating the impressive accuracy that can be achieved with a limited number of replicas. In Section IV we show another example, namely the ferromagnetic Heisenberg Model Suzuki (1976); Barma and Shastry (1978), once again comparing the exact quantum results with probabilistic simulations of the p-bit network.
Finally in Section V we present SPICE simulations of actual hardware implementations that can be built with existing Embedded Magnetoresistive RAM (eMRAM) technology that has been under development by a number of foundries Lin et al. (2009); Song et al. (2016); Shum et al. (2017). Unlike standard eMRAM where a non-volatile MTJ is carefully engineered with a large energy barrier (-) so that the magnetization state is retained for a long time Bhatti et al. (2017), the free layer of the MTJ for the p-bit is designed as a thermally unstable magnet ( whose magnetization rapidly fluctuates in time in the presence of thermal noise Camsari et al. (2017b). Using full device-level SPICE simulations corresponding to the p-bit and a resistive interconnection matrix, we demonstrate that the correct quantum correlations can be obtained using this classical p-circuit which can be built with existing technology at room temperature. With appropriate magnet designs Hassan et al. (2019) individual p-bits can flip in a nanosecond or less so that with a million of them operating in parallel, we should have petaflips per second which is quite impressive compared to existing digital implementations Fang et al. (2014).
Ii q-bit to p-bit
Since the seminal work of Suzuki Suzuki (1976), it is well-known that a -dimensional quantum many-body Hamiltonian can be mapped to a +1-dimensional classical Hamiltonian applying the so-called Suzuki-Trotter decomposition Trotter (1959); Suzuki (1976), which is used as a basis for PIMC methods to simulate quantum annealing using classical computers Santoro et al. (2002). This decomposition results in the quantum system being mapped to a classical system with replicas that are coupled to each other. In this paper we consider two examples as described in the next two Sections, but the principles apply to stoquastic Hamiltonians in general.
Consider for example the Transverse Ising Hamiltonian in 1D written as Pfeuty (1970):
The Suzuki-Trotter mapping produces the following classical 2D Hamiltonian Santoro et al. (2002):
where , being the number of replicas, and the vertical coupling term is and . Note how the quantum mechanical operators in Eq. 1 have become classical spins in Eq. 2. The mapping of Eq. 2 becomes exact in the limit of infinite replicas () however, for finite replicas the error scales as Heim et al. (2015) and can be made arbitarily small by choosing an appropriate number of replicas.
Behavioral model for p-bits
|where is dimensionless time that is incremented one at a time, is a random number uniformly distributed between 1 and +1 and at each time step is uncorrelated with the chosen at the previous step. is the dimensionless current to each p-bit, where is the inverse temperature. in general, is calculated according to,|
|which in the present case, becomes:|
where is the interconnection matrix and is the bias term. We refer to Eq. 3 as Probabilistic Spin Logic (PSL) equations and note that these equations are essentially the same as those discussed in the context of stochastic neural networks such as Boltzmann Machines, developed by Hinton and colleagues Ackley et al. (1985).
It is important to note that while Eq. 3b is a linear synapse that typically arises from quadratic Hamiltonians with 2-body interactions, specially designed digital CMOS circuits can be used to implement more complicated interactions arising from cost functions such as generalized Hopfield models with -body interactions Gardner (1987); Seki and Nishimori (2015). Such a flexibility of implementing complicated interactions could be a key advantage for hardware p-circuits.
PSL equations can be updated to approximate the steady state joint probability density for any matrix, symmetric or asymmetric. For symmetric matrices, the joint probability density is simply expressed by the classical Boltzmann Law, , where is the energy for a given configuration , . There are two important conditions regarding the updating of Eq. 3. First, Eq. 3b needs to be calculated much faster than Eq. 3a for proper convergence Pervaiz et al. (2017a), a requirement particularly relevant for hardware implementations. Second, Eq. 3a needs to be updated sequentially, as in Gibbs sampling Geman and Geman (1984). The requirement of sequential updating prohibits parallelization in software implementations, except in special cases such as restricted Boltzmann machines where the lack of intralayer connections between “visible” and “hidden” layers allows each layer to be updated in parallel Hinton (2012). For asynchronous hardware implementations, however, a clockless operation seems to satisfy the requirement of sequential updating naturally Pervaiz et al. (2017a); Sutton et al. (2017).
Iii Transverse Ising Hamiltonian
For the 1D Transverse Ising Hamiltonian (Eq. 1), we assume periodic boundary conditions such that . is the local transverse magnetic field and is a local -directed magnetic field. Eq. 1 can be constructed by first writing each term, , as a matrix followed by ordinary matrix multiplication for each product term. These terms are written in terms of Pauli spin matrices () at the lattice point as where is the 2 identity matrix and is the Pauli spin matrix at the term in the product.
Quantum Boltzmann Law
In principle, Eq. 1 can be exactly solved for any quantity of interest as a function of temperature and all other parameters and , from the principles of quantum statistical mechanics Kadanoff and Baym (1962):
where is the “inverse temperature” (as defined in Eq. 3a) and we have chosen to use a unit system in which . is the quantity we wish to calculate with a corresponding operator . In practice, directly solving Eq. 4 becomes intractable due to the exponential dependence of the Hamiltonian () to the size of the problem (). Due to its similarity to the classical Boltzmann Law Feynman et al. (2011), we refer to Eq. 4 as the “Quantum Boltzmann Law” throughout this paper and solve it for small 1D systems. To obtain numerically stable results at low temperatures (high ), we first diagonalize the Hamiltonian and subtract the ground state energy from the diagonals, without changing any observable quantities.
Averages and correlations
In FIG. 2a we calculate the average -magnetization of a 1D ferromagnetic () chain with spins, as a function of a transverse magnetic field. The average -magnetization, , is obtained by the operator where provides the net -spin, , at site . To break the symmetry of at low temperatures we introduce a -directed magnetic field. As the transverse magnetic field increases, gradually decreases, while (not shown) increases, as spins become aligned with the transverse magnetic field. Incidentally, the reverse process, starting from a large at a low temperature and slowly decreasing it to find the ground state of a complicated spin-glass, is commonly used in quantum annealing algorithms Heim et al. (2015).
FIG. 2b shows the probabilities of correlated states at a given temperature and transverse field expressed as decimal numbers. This is done by first converting the states to binary numbers such that denotes +1 and denotes 0 and then converting the full state into a decimal number, for example the all down state corresponds to 0, and the all up state corresponds to 255 and so on. There are such states, each with a given probability obtained from Eq. 4. These correlated states are calculated by first constructing an operator for the probability of finding a state at a given site, where is the identity matrix. Similarly, . Using these operators, any correlation of the form can be calculated from the corresponding composite operator:
There are 256 such operators and Eq. 4 can be used for each of them to obtain a probability for each state for any . FIG. 2b shows these probabilities at a chosen parameter combination and they are in agreement with results obtained from a simulation of p-bits, as we next explain in Section II. Note that this joint probability density contains all statistical information in the system, as averages and other correlations of interest can be calculated from it, for example one can obtain by weighting each state by the net -spin they contribute to the average.
PSL vs Quantum Boltzmann Law
With this picture, the mapped classical Hamiltonian with replicas described in Eq. 2 is used to obtain a consolidated matrix that is of size to be used in Eq. 3. FIG. 2 shows the equivalence of the PSL implementation of the Transverse Ising Hamiltonian to the exact quantum many-body description for a 1D-chain with spins. Note that the p-bit mapping can be applied to much larger spin systems, but an exact solution by Eq. 4 quickly becomes intractable. We investigate the average -spin of this ferromagnetic chain at a constant temperature () as a function of the transverse magnetic field, . A symmetry breaking field (to favor a +1 order) of is applied. As expected, the exact result shows how the average -spin becomes disordered. The PSL results for a replica system reproduce this behavior. The -spin average is obtained by taking an average over the length of the chain, as well as over each replica. The final average (for a given red dot) is recorded at the end of dimensionless time steps. Since a single stochastic point is recorded at the end of , for each point, we observe a variance in the final results, however averaging over 100 different simulations for the same system, we get a very close match to the exact solution.
In FIG. 2b, the full joint probability density for the classical system is obtained from a PSL simulation that is run for dimensionless time steps. The state of each replica with 8-spins is converted into a binary number at each time step, as in the exact solution, and then collected over all replicas. The striking agreement with PSL and the Quantum Boltzmann Law in FIG. 2 establishes the faithful mapping of the quantum system to the classical system, from the behavioral PSL equations Eq. 3.
Iv Ferromagnetic Heisenberg Model
Before proceeding to a hardware implementation showing how replicated networks of p-bits can be built by existing nanodevices, we show another example of a stoquastic Hamiltonian that can be represented by p-bits. The Heisenberg Hamiltonian in 1D in the presence of a transverse magnetic field can be written as:
Following Suzuki (1976); Bravyi (2014); Barma and Shastry (1978), we apply the Suzuki-Trotter transformation to this system and obtain the chessboard lattice that is shown in Fig. 3b with shaded and unshaded unit cells. The interactions terms for this hardware neural network are shown in Fig. 3c. For the shaded unit cells all two-body interactions () exist in addition to a 4-body interaction () that involves the product of each spin. The two-body interactions can be implemented using a linear synapse of the form of Eq. 3b but the 4-body interaction requires a non-linear synapse that computes the input terms that are products of three neighboring spins. The interaction terms arise due to the diagonal parts of the quantum system, as in the case of the Transverse Ising Hamiltonian, and exists for both the shaded and unshaded unit cells shown in Fig. 3c. The detailed derivations of all interaction terms are shown in Appendix A.
In Fig. 3d, we show a simulation of the classical system using behavioral PSL equations and compare this to the exact solution as before. We choose a set of parameters, that corresponds to the ferromagnetic Heisenberg Model with a small transverse magnetic field in the -direction, such that all off-diagonal terms in the are positive, hence making this system stoquastic Bravyi (2014). In this small example with spins, we observe good agreement between the mapped system and the exact solution.
V p-bit to Stochastic MRAM
We now show how the behavioral p-bit model can be represented by a stochastic neural network in hardware (FIG. 4a). Each replica in the classical system consists of p-bits (neurons) that are interconnected to each other with a resistive network (synapse), a typical architecture often used in many hardware neural networks Yu et al. (2011); Hu et al. (2016) though for more complicated systems involving -body interactions (), standard electronic devices such as FPGA’s could also be used for this purpose, for example as in Ref. McMahon et al. (2016). The extra dimension added by the Suzuki-Trotter transformation would increase the synaptic complexity but for sparse quantum networks, this transformation would only slightly increase the fan-in of each p-bit in the classical network.
We assume that the weighted summation is carried out by ideal operational amplifiers. The replicas are also connected in the vertical direction (not shown in FIG. 4) with nearest neighbor coupling according to the coupling coefficient .
In the case of quantum annealing, the vertical resistors need to be reconfigurable, therefore they need to be designed differently compared to the fixed resistors that represent the transverse coupling . In our device level examples, we use fixed resistors in order to establish the equivalence between the classical and quantum systems and have not performed annealing.
where is a unit resistor that is used to electrically change the inverse temperature , and is a transistor dependent parameter ( mV) that defines the stochastic window of the p-bit (FIG. 4c). Depending on the sign of the interconnection, , the non-inverted output or the inverted output is used for the synaptic connections.
The 1T/1MTJ p-bit is modeled by combining a 14 nm-High Performance FinFET model from the open source Predictive Technology Models (PTM) pre () with a stochastic Landau-Lifshitz-Gilbert (sLLG) solver implemented in SPICE Torunbalci et al. (2018), following the design described in Camsari et al. (2017b) (FIG. 4d). The MTJ is modeled as a simple conductor whose conductance depends on the instantaneous magnetization , provided by the sLLG such that
where and are the parallel and antiparallel resistance of the MTJ and . We use an experimentally measured value for the tunneling magnetoresistance (TMR) after Ref. Lin et al. (2009). is set equal to the transistor resistance at to produce a symmetric sigmoid with no offsets, in this case . The free layer is assumed to be a circular low barrier nanomagnet Cowburn et al. (1999); Debashis et al. (2018) with a diameter of 22 nm and thickness of 2 nm and a saturation magnetization of emu/cc, with a damping coefficient , typical parameters for CoFeB Sankey et al. (2008).
The time dependent magnetization is obtained by solving sLLG equation in the monodomain approximation Sun (2000):
is the electron gyromagnetic ratio, is electron charge and is the number of Bohr magnetons () in the volume of the magnet, . contains the external magnetic and internal anisotropy fields of the magnet as well as the noise field. In the case of a circular nanomagnet without an easy-axis anisotropy, the total internal magnetic anisotropy becomes , where - is the easy plane of the magnet. The thermal noise is added in three directions (, and ) with zero mean and in units  Li and Zhang (2004).
The p-bit shown in FIG. 4c is a series-resistance controlled device where the transistor resistance can be made much smaller or much larger compared to the fluctuating MTJ resistance. Therefore, the operation of the p-bit does not require manipulating the magnetization of the free layer unlike in standard spin-transfer-torque MRAM cells. However, the current flowing through the fixed layer of the MTJ produces a spin-polarized spin current that can unintentionally torque the magnet. We assume that this current is given as , where is the fixed layer orientation and is an interface polarization that can be related to TMR Datta et al. (2012). This spin-current is fed back to the sLLG solver and fully accounted for in the calculation of magnetization in our simulations, however for the circular LBM with a large demagnetization field used here, its effects are negligible Faria et al. (2017). Using these models, FIG. 4c shows transient SPICE simulations of a single p-bit output, for 1000 samples where is rapidly swept in 2 ns. The range of stochastic outputs is bounded by a distribution of resistances ranging from to . The ensemble average shows an approximate hyperbolic tangent behavior that allows the mapping shown in Eq. 7.
The inset of FIG. 4d shows the autocorrelation time of the circular in-plane magnet with a lifetime of ps. The fluctuations for a circular magnet is expected to be faster compared to a magnet with perpendicular anisotropy due to the strong demagnetizing field that keeps the magnetization vector in the easy plane of the magnet Lopez-Diaz et al. (2002). The very short lifetime of such a circular low barrier magnet could allow very fast and efficient sampling times, as long as the interconnection network operates faster than these timescales. In present simulations, the resistive network operates instantaneously with an ideal operational amplifier therefore this requirement is met naturally, however in real implementations the synapse needs to be designed carefully.
The second requirement, the need for sequential updating of each p-bit is met naturally since the probability of simultaneous flips among p-bits is extremely unlikely, therefore hardware p-bits go through an effectively random update order that does not affect their final distribution.
Stochastic MRAM-based p-bit vs Quantum Boltzmann Law
In FIG. 5, using full SPICE simulations for a 40 p-bit network we compute the joint probability density of a spin ferromagnetic chain () using 10 replicas, with and . Unlike FIG. 2, no symmetry breaking field is applied and the network is asynchronously operated for ns, with a time step of 1 ps. All analog voltage values at the end of the SPICE simulation are thresholded (, ) and a time-average is obtained similar to the PSL averaging after converting the state of each p-bit to binary and then to decimal. The results from the full device models seem to be in good agreement with the exact solution obtained from Eq. 4 and the behavioral PSL equations that are included for reference. Note the suppression of states and that correspond to the energetically unfavorable antiferromagnetic configurations and , respectively. The agreement between the full SPICE models with the behavioral and exact solutions establishes the feasibility of the proposed quantum circuit emulator.
We have presented a scalable, room-temperature quantum emulator using stochastic p-bits that can be built by a simple modification of the existing 1T/1MTJ cell of the eMRAM technology. The proposed emulator uses physical replicas for repeated Trotter slices used in software Quantum Monte Carlo methods. Having physical replicas for each slice could enable better scaling properties for quantum annealing compared to classical annealing as discussed in Santoro et al. (2002), since choosing the optimal number of replicas or probing each replica separately to find better energy minima is possible in a physically engineered design, unlike in real quantum systems Heim et al. (2015). The electrical control of annealing parameters, inverse temperature () and transverse field (), could allow a very large number of q-bits to be reliably emulated with room temperature p-bits. Using conventional electronic devices such as GPU’s or FPGA’s to implement the synapses, it should be possible to engineer complicated interactions that extend beyond nearest neighbors and/or involve -body interactions (). We note that even though the “sign problem” limits the universal use of our p-computer, a large number of practically relevant quantum systems could be efficiently emulated by it, considering a large number of optimization problems have been mapped on to the Transverse Ising Hamiltonian Lucas (2014). Our results provide a method of emulating quantum systems with probabilistic hardware in advance of a scalable universal quantum computer.
Acknowledgements.KYC is grateful to Brian M. Sutton for helpful discussions. This work was supported in part by ASCENT, one of six centers in JUMP, a Semiconductor Research Corporation (SRC) program sponsored by DARPA.
Appendix A : Mapping Quantum Heisenberg Model to a Classical System
The Heisenberg Hamiltonian emulated in Section IV is,
The -th approximant of the Suzuki-Trotter transformation for this Hamiltonian is then given by
and the classical system becomes,
with periodicity along ( dimension such that . Next, we evaluate and , with each being a by 1 vector and , where denotes spin of replica. With this choice of and and using some identities, these terms can be decoupled and obtained by evaluating the 44 density matrix:
where, , are 44 Pauli spin matrices. The corresponding are given by:
We then use the following relation:
where is a constant that we ignore and
This corresponds to the energy model for the Heisenberg Hamiltonian as shown in Fig. 3.
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