Incoherent approximations of leakage for efficient simulations of noisy quantum computations

Abstract

Methods, systems and apparatus for determining an effect of uncontrolled quantum state transitions in a noisy quantum computation performed by a quantum computing device, where the noisy quantum computation comprises execution of a quantum circuit that is represented by one or more quantum channels. In one aspect, the uncontrolled quantum state transitions are approximated as incoherent uncontrolled quantum state transitions through application of a random phase approximation to the one or more quantum channels to obtain a modified quantum circuit that is represented by one or more incoherent quantum channels. The incoherent quantum channels preserve incoherence between computational subspaces and environmental subspaces for the one or more quantum channels. A simulation of the modified quantum circuit is then performed using a qubit simulation of the modified quantum circuit.

Description

[0001]Attorney Docket No. 56113-0504WO1 INCOHERENT APPROXIMATIONS OF LEAKAGE FOR EFFICIENT SIMULATIONS OF NOISY QUANTUM COMPUTATIONS BACKGROUND This specification relates to quantum computing. Quantum error correction provides a means for implementing quantum computation in a fault-tolerant manner, despite the presence of unavoidable physical noise. This is demonstrated by threshold theorems, which show that as long as the physical error rate is below some threshold value, logical errors can be exponentially suppressed in the size (distance) of a code. This then guarantees that, with high probability, logical information can be stored and manipulated for times sufficient to be useful quantum computation. The threshold theorems make certain assumptions on the noise, and when these are violated, often a sharp decrease in code performance results. This is particularly true for non-computational errors, where degrees of freedom transition to physical states outside of the (typically two-level) computational subspace. Non-computational errors occur in most quantum computing architectures, e.g., due to loss in photonic and trapped ion settings or excitations out of the qubit subspace in superconducting qubit architectures. A superconducting qubit is typically modeled as an anharmonic oscillator, where the lowest two (qubit) levels span the computational subspace and excited states are non-computational states. The process of exciting states outside of the computational subspace is known as “leakage”. Though great care is taken to avoid such transitions, they are inevitable with current technology, and can arise from simple heating mechanisms, or more complex coherent transitions. Since leakage errors are out of the scope of most current error correcting codes they can greatly reduce code performance. For example, an effective distance reduction occurs in the repetition code under these errors. In addition to posing a challenge to error correction, leakage also dramatically increases the cost of classical and quantum simulation. While there do exist efficient classical simulators of many error correcting codes relying on the stabilizer representation that can accommodate thousands of qubits, certain noise models (such as leakage) do not fit naturally into this framework. Thus, there remains a great demand for fully quantum simulations for use in verification and validation. Attorney Docket No. 56113-0504WO1 In addition to leakage, there are other physical error mechanisms occurring in physical implementations of quantum computing which require non-Markovian descriptions. Unlike Markovian errors, non-Markovian errors cannot be described using just the current state of the system at a given time - rather, some notion of environmental memory (or a history of the errors) must be included. In this context, it can be beneficial (in terms of computing resource overhead) to use approximate descriptions for these errors, specifically those that do not require a full quantum-mechanical model. SUMMARY This disclosure describes techniques for incoherent approximations of leakage and other non-Markovian sources of error in simulations of noisy quantum systems. In general, one innovative aspect of the subject matter described in this specification can be implemented in a method performed by a quantum computing device, the method comprising: determining an effect of uncontrolled quantum state transitions in a noisy quantum computation performed by the quantum computing device, wherein the noisy quantum computation comprises execution of a quantum circuit that is represented by one or more quantum channels, the determining comprising: approximating the uncontrolled quantum state transitions as incoherent uncontrolled quantum state transitions, comprising applying a random phase approximation to the one or more quantum channels to obtain a modified quantum circuit that is represented by one or more incoherent quantum channels, wherein the incoherent quantum channels preserve incoherence between computational subspaces and non-computational subspaces for the one or more quantum channels; and performing a simulation of the modified quantum circuit, comprising performing a qubit simulation of the modified quantum circuit. Other implementations of these aspects includes corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination thereof installed on the system that in operation causes or cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions. Attorney Docket No. 56113-0504WO1 The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations the simulation comprises a discrete time quantum trajectory simulation. In some implementations performing the discrete time quantum trajectory simulation comprises, for each incoherent quantum channel: iteratively evolving a quantum-classical state of a system of qubits and a classical register included in the quantum computing device from an initial quantum-classical state to a final quantum-classical state, wherein the classical register records Hilbert space subspaces occupied by the qubits in the system of qubits during the evolving. In some implementations the modified quantum circuit is represented by a composition of the incoherent quantum channels and wherein a final quantum-classical state obtained for a preceding incoherent quantum channel in the composition of incoherent quantum channels is provided as an initial quantum-classical state for a subsequent incoherent quantum channel in the composition. In some implementations iteratively evolving comprises, for each iteration: identifying, by classical computation, a current Hilbert space subspace occupied by quantum degrees of freedom of the quantum-classical state for the iteration, comprising performing a lookup in the classical register; sampling, by classical computation, terms of reduced Kraus operators that act on the current Hilbert space subspace; applying, by quantum computation, the sampled terms of the reduced Kraus operators to the quantum degrees of freedom of the state for the iteration; and updating, by classical computation, the classical register based on the sampled terms of the reduced Kraus operators. In some implementations the random phase approximation comprises an average over a uniform distribution of distinct random phases of: a conjugate of the quantum channel with respect to a phase unitary operator, wherein the phase unitary operator comprises an operator that assigns a distinct random phase to each subspace in a decomposition of a Hilbert space for the quantum circuit into a direct sum of subspaces; and a conjugate of the quantum channel with respect to an inverse of the phase unitary operator. In some implementations the one or more quantum channels are represented by respective Kraus operators that act on a full Hilbert space for the simulation; and applying the random phase approximation to the one or more quantum channels comprises generating reduced Kraus operators that act on subspaces of the full Hilbert space. Attorney Docket No. 56113-0504WO1 In some implementations the reduced Kraus operators comprise off-diagonal terms associated with incoherent transitions, wherein each off-diagonal term comprises a projection operator onto a first subspace of the full Hilbert space, a Kraus operator that acts on the full Hilbert space, and a projection operator onto a second subspace of the Hilbert space, wherein the second subspace is different from the first subspace. In some implementations the reduced Kraus operators comprise block diagonal terms that preserve Hilbert space subspaces, wherein each block diagonal term comprises a sum, over one or more subspaces of the Hilbert space, of operators comprising a projection operator onto the subspace, a Kraus operator that acts on the full Hilbert space, and the projection operator onto the subspace. In some implementations the reduced Kraus operators comprise projection operators onto respective subspaces of the full Hilbert space, wherein the number of subspaces is dependent on a type of uncontrolled quantum state transitions. In some implementations the method further comprises determining a Hilbert space decomposition for the uncontrolled quantum state transitions in the noisy quantum computation; and determining projection operators for each subspace in the Hilbert space decomposition, wherein the projection operators define reduced Kraus operators for the simulation. In some implementations uncontrolled quantum state transitions comprise transitions of a single qudit out of the computational subspace and into a leakage subspace, and wherein the Hilbert space decomposition comprises a first subspace spanned by computational quantum states 0 and 1, and a second subspace spanned by a leakage quantum state 2. In some implementations the uncontrolled quantum state transitions comprise transitions of n qudits out of the computational subspace and into a leakage subspace, wherein the Hilbert space decomposition comprises 2nsubspaces. In some implementations the uncontrolled quantum state transitions comprise coupler state transitions or transitions of a two-level system (TLS), and wherein the subspaces of the Hilbert space comprise a first subspace corresponding the coupler or TLS being in a zero state and a second subspace corresponding to the coupler or TLS being in a one state. In some implementations the method further comprises processing results of the simulation to obtain simulation statistics that represent the effect of the uncontrolled quantum state transitions in the noisy quantum computation performed by the quantum computing device. Attorney Docket No. 56113-0504WO1 In some implementations performing the qubit simulation of the modified quantum circuit comprises simulating single Hilbert space subspaces. In some implementations the quantum circuit comprises a quantum error correction circuit. In some implementations the uncontrolled quantum state transitions comprise interactions between the quantum computing device and an environment. The interaction can include an interaction with one or more environmental modes. The environmental modes can include modes of a readout resonator or a strongly coupled two level system. The interaction can be a strong interaction. The interaction can be coherent leakage and the non- computational subspaces can be leakage subspaces. Performing the simulation of the incoherent interaction can include simulating leakage caused by incoherent heating processes or leakage caused by coherent transitions during entangling operations in the quantum circuit. The interaction can represent non-Markovian errors. In some implementations performing the discrete time quantum trajectory simulation comprises iteratively evolving a quantum state of a system of qubits in a quantum computer from an initial quantum state to a final quantum state, wherein iteratively evolving comprises, for each iteration: identifying, by classical computation, a current Hilbert space subspace occupied by the quantum state for the iteration, comprising performing a lookup in a plurality of classical registers that each record a Hilbert space subspace occupied by a respective qubit in the system of qubits; sampling, by classical computation, terms of modified Kraus operators that correspond to the current Hilbert space subspace; applying, by quantum computation, the sampled terms to the quantum state for the iteration; and updating, by classical computation, the plurality of classical registers based on the sampled terms. In some implementations the initial quantum state comprises a pure quantum state and Kraus operators that represent the plurality of quantum channels act on O(1) qubits in the system of qubits. The Kraus operators can be non-square matrices. In some implementations performing the qubit simulation of the modified quantum circuit comprises simulating single Hilbert space subspaces. In some implementations the simulation comprises a density matrix simulation or continuous time simulation. The subject matter described in this specification can be implemented in particular embodiments so as to realize one or more of the following advantages. Attorney Docket No. 56113-0504WO1 Performing computations to simulate quantum systems is computationally expensive – the cost grows exponentially in the number of quantum degrees of freedom. Including error mechanisms in simulations to make them more accurate and provide information about the performance of the quantum computer requires the introduction of even more quantum degrees of freedom. In practice, the simulation cost scales exponentially in the number of degrees of freedom. This means that the feasible size of a noisy simulation (a simulation that includes error mechanisms) is smaller than the size of a noiseless simulation (a simulation that does not include error mechanisms). In many applications, it is important to understand how a noisy quantum system will behave. However, many of these applications are too large to be computationally feasible. A system implementing the presently described techniques can achieve improved computational efficiency, e.g., in terms of required memory and time overhead, when simulating noisy quantum systems. For example, in the case of leakage errors, the memory requirements for a standard simulation of ^^ three-level systems scales as ^^^3^^ compared to ^^^2^^ in the qubit case. Given a fixed computational budget, this effectively reduces the achievable system size ^^. However, by applying examples of the presently described techniques where leakage (or other uncontrolled quantum state transitions) is approximated as an incoherent process, simulations with leakage (3 or 4 levels per qudit) can be performed only using the resources of a qubit (2 level) system. In other words, the presently described techniques allow for the simulation of leakage without significantly increasing memory requirements. In addition, standard implementations of discrete time quantum trajectory simulations scale as ^^^^^ଶ^. However, examples of the presently described techniques use pure initial states and Kraus operators with bounded locality (i.e., act only on O(1) qubits). Therefore, the memory and time requirement to sample a single trajectory is reduced to O(N). This scaling is essential to allow for simulations of, e.g., a distance 5 rotated surface code. The details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims. BRIEF DESCRIPTION OF THE DRAWINGS FIG.1 depicts an example computing system. Attorney Docket No. 56113-0504WO1 FIG.2 is a diagram of the dynamics of a quantum-classical state vector in an iteration of a quantum trajectory simulation performed by the quantum computing system of FIG.1. FIG.3 is a flow diagram of a first example process for determining an effect of uncontrolled quantum state transitions in a noisy quantum computation performed by a quantum computing device. FIG.4 is a flow diagram of a second example process for determining an effect of uncontrolled quantum state transitions in a noisy quantum computation performed by a quantum computing device. DETAILED DESCRIPTION Uncontrolled quantum state transitions during a quantum computation can introduce errors into the quantum computation. For example, in quantum error correcting codes, uncontrolled quantum state transitions out of the computational subspace and into the leakage subspace cause errors that can last for multiple detection rounds and significantly contribute to logical errors. It is therefore important to understand how to model and simulate the effect of uncontrolled quantum state transitions in a quantum computation performed by a quantum computing device, e.g., to determine whether dedicated error-mitigating operations such as quantum gates that remove leakage should be included in the quantum computation. Fully quantum simulations of uncontrolled quantum state transitions require additional quantum degrees of freedom, e.g., qubit levels, which substantially limits the system sizes that may be simulated. This specification describes techniques for simulations of uncontrolled quantum state transitions. The techniques include a Random Phase Approximation (RPA) for quantum channels that preserves the incoherence between the computational and leakage subspaces. The assumption of incoherence enables the simulation of uncontrolled quantum state transitions using only the resources required of a qubit system and a classical register subsystem. The techniques described herein are primarily described with reference to simulating leakage (e.g., uncontrolled quantum state transitions of a single qudit out of the computational subspace and into a leakage subspace). The leakage can be incoherent leakage, e.g., leakage caused by incoherent heating processes, or coherent leakage, e.g., leakage caused by coherent transitions during entangling operations in the quantum error correction circuit. However, the techniques can be applied to other settings and device components where uncontrolled quantum state transitions occur, e.g., to simulate the effect of Attorney Docket No. 56113-0504WO1 uncontrolled coupler state transitions, uncontrolled transitions of readout resonator states, or uncontrolled transitions of two-level systems (TLS). Generally, the techniques can be applied to any quantum system with a subsystem whose Hilbert space can be decomposed as a sum of individual subspaces and for which a mechanisms exists that causes the state of the subsystem to decohere with respect to those subspaces. FIG.1 depicts an example computing system 100. The example computing system 100 is an example of a system implemented as part of a hybrid classical-quantum computing device in which the systems, components and techniques described in this specification can be implemented. The computing system 100 includes a comprehensive device simulation system 102 and a device design module 104. In some implementations the system 100 can include a quantum computing device 110. In some implementations, some or all of the components of the example computing system 100 can be directly connected. In other implementations, some or all of the components of the example computing system 100 can be connected through a network, e.g., a local area network (LAN), wide area network (WAN), the Internet, or a combination thereof. The comprehensive device simulation system 102 and device design module 104 can be implemented as one or more computer programs, i.e., one or more modules of computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, a data processing apparatus. The computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of them. The comprehensive device simulation system 102 and device design module 104 are configured to perform operations for determining an effect of uncontrolled quantum state transitions in a noisy quantum computation performed by a quantum computing device, e.g., quantum computing device 110. A quantum computation is a quantum operation (or sequence of operations) that is applied to a quantum state of a quantum system. The quantum operation can be described as a completely-positive and trace preserving (CPTP) map that acts on a density matrix for the quantum system. Such maps are referred to as quantum channels. A quantum channel ℰ can be represented through its action on a density matrix ρ, Attorney Docket No. 56113-0504WO1 ℰ^^^^ ൌ ^ ^^ற ^^^^^^^1^ ^ஹ^ where in Eq. (1), the set of operators, which satisfyidentity ∑^ஹ^ ^^ற^ ^^^ ൌ ^^ to preserve the trace of ρ. A noisy quantum circuit (that represents aquantum computing program / simulation) can be realized as a sequence of such quantumchannels, ℰ ൌ ℰ^ ◦ ^ ^ ^ ◦ ℰ^. In some implementations, a single noisy operation (such asevolution during a gate) can be represented as a composition ℰீ ൌ ℰ′ ◦ ℰ^ of ℰ^, where ℰ^denotes the ideal target unitary (single Kraus operator ^^^ ൌ ^^) and ℰ′ takes into account arelevant error mechanism (e.g., dissipation or leakage). In the limit where multiple weak error mechanisms are present (e.g., phase errors, depolarizing errors, or amplitude damping) multiple channels ℰ′ can be composed after ℰ^. For a system with exponentially largedimension N (e.g. ^^ ൌ 2^ for n qubits), a naive simulation of the channel on an arbitraryinitial density matrix ρ would require multiple matrix-matrix multiplications. Since the matrices above have dimension N × N, the space required will scale at least as ^^^^^ଶ^ and the total number of operations will scale as ^^^^^ଷ^. The comprehensive device simulation system 102 is configured to simulate a noisy quantum computation (i.e., a noisy quantum circuit). The simulation captures the effects of uncontrolled quantum state transitions, e.g., leakage out of the computational subspace, on the performance of the quantum computing device 110 when the quantum computing device 110 executes the quantum computation. For example, the simulation can be used to determine how sensitive the failure rate (e.g., logical error probability) of the quantum computation is to uncontrolled quantum state transitions. This information can be used to determine whether additional dedicated operations, e.g., gates, need to be added to the quantum circuit to counteract the uncontrolled quantum state transitions, e.g., remove leakage. The simulation performed by the comprehensive device simulation system 102 can be a discrete time quantum trajectories method. The discrete time quantum trajectories method is a technique for sampling the output of a sequence of quantum channels. Each trajectory of multiple trajectories represents an independent experimental simulation of a pure state. To simulate a trajectory, a pure initial quantum state |ψ^ is assumed. As described above, evolution of a noisy quantum circuit can be described as a sequence of quantum channels, Attorney Docket No. 56113-0504WO1 each composed of respective Kraus operators. Instead of applying a given channel to a density matrix, one of its Kraus operators ^^^is sampled using, e.g., the Born Rule probability ଶ ^^^^^^ ൌ ^ห^^^ |^^^ห^ ൌ ^^^|^^ற^ ^^^|^^^. (2) and applied to the state ^ ^^^^^ ^^^|^^^. The random sampling and state update is identical to the backaction of a positiveoperator-valued measure (POVM) with elements ^^^ ൌ ^^ற ^ ^^^. As such, at end of eachtrajectory the record ^^^, ^^ଶ, ... of all sampled Kraus operators is obtained, including anymeasurement outcomes. As described in more detail below, in examples of the presently described the state vector (and corresponding Hilbert space) can dynamically change size. A quantum channel that changes the Hilbert space dimension will have Kraus operators described by non-square matrices. The quantum channels description can be generalized to include classical registers. Each classical register can be formally represented as an independent subsystem. The defining property of a classical subsystem is that the state vector is always incoherent with respect to a fixed basis ^|^^^^. For a quantum trajectory, this means that the subsystem always remains in a product state. For the case of classical registers, this means that the state vector is always supported by exactly one basis operator. In other words, at any given time in the trajectories simulation, exactly one observable of the set ^|^^^^^^|^ has non-vanishing expectation value. Since the full system will always be of tensor product form |^^^ௌ ⊗ |^^^^ ⊗ |^^ଶ^ ... ^3^(where the left hand side |^^^ௌrepresents the quantum degrees of freedom and the right hand side represents the classical degrees of freedom) the cost of including classical registers is linear (instead of exponential) in the number of classical subsystems. Channels acting on both classical and quantum degrees of freedom can be expressed in terms of Kraus operators of the form Attorney Docket No. 56113-0504WO1 ^^^^^^ ⊗ |^^^^^^|, ^4^where ^^^^^^ acts on the quantum degrees of freedom and |r^^r| on the classical degrees of freedom, and r represent a multi-index of all register values. In practice most channels will only depend on at most one register’s value. It can be verified that the trace preservation condition is satisfied by Eq. (4). The operations given by Eq. (4) act by applying the quantum channel ^^^^^^^ ^ to the quantum degrees of freedom, conditioned on the current ^ the classical registers in state r. In order to preserve incoherence between classical registers, the Kraus operators given by Eq. (4) must be diagonal in the fixed basis of each register. Like quantum degrees of freedom, classical registers are allowed to be created or destroyed. The Kraus operators representing these processes take the same form as in thequantum case described above. One generalization is Kraus operators of the form ^^^ ⊗ |^^^,where ^^^acts on the quantum degrees of freedom and |^^^ denotes the value of a new register. These Kraus operators can be used to model the recording of a sampling outcome (i.e., which index j was sampled) into a classical register. An example is the recording of a POVM^^^^^^, which can be represented using Kraus operators ^^^^ ⊗ |^^^.In addition to the recording of sampling into registers, the simulations performed the comprehensive device simulation system 102 use classical operations which act only on classical registers. Unlike the classically conditioned quantum channels described above, these channels are able to change the register states. Such operations can be modeled as the probabilistic application of function ^^^with probability ^^^on the classical registers using Kraus operators of the form ^^^,^ ൌ √^^^ ∑௧̅∈^^^|^^^^^^̅^^^^^̅|(5) where for each ^^, the input space of ^^^^^^ has been partitioned into disjoint subsets ^^^. This partitioning can be chosen such that ^^^is injective when restricted to a given subset of inputs ^^^^^^ , which guarantees that the resulting Kraus operators are trace preserving. Because ^^^is Attorney Docket No. 56113-0504WO1 injective on the subset ^^^^^^ , the above Kraus operators can never generate superpositions of classical register The Hilbert for the simulation can be decomposed into a direct sum ofsubspaces. The dephasing channel for the subspaces ℋ^ୀ^,…ௌ is given by Δ^^^^ ൌ∑ௌ ^ୀ^ ^^^^^^^^where ^^^is a projection operator onto subspace ℋ^and ^^ is a quantum state. This channel is equivalent to measuring which subspace a given state is in and discarding that information; it destroys coherence between states in different subspaces. For example, starting with a pure state that is a coherent superposition of states from different subspaces, the action of the dephasing channel Δ produces a mixture of the states. Accordingly, the quantum state ^^ is incoherent with respect to the subspaces ℋ^ୀ^,…ௌif it is invariance underthe dephasing channel, i.e., if ^^ ൌ Δ^^^^. A quantum state is incoherent if and only if it is astatistical mixture of states each contained in a (possibly different) single subspace ℋ^. This is equivalent to saying that it is block diagonal with respect to the subspaces ^^^. generally, any operator A is defined as incoherent with respect to subspaces ^^^if ൌ^^^^^^ ൌ ∑ௌ ^ୀ^ ^^^ ^^^^^. A quantum channel is incoherent if it admits a set of Kraus operators^^^^^^such that,for all ^^^ and any incoherent state ^^ ൌ Δ^^^^,^^^^^^ற ൌ ^^^ற^^ ^^^^^^^^^ . (6) Such a set of Kraus operators has the property that, for any ^^^and subspace projector ^^^,there exists another projector ^^^ such that ^^^^^^ ൌ ^^^^^^^^^. In the context of a trajectories simulation, this means that a state which starts incoherent will remain incoherent, irrespective of which Kraus operator was sampled. It is noted that a given channel can have more than one distinct Kraus operator representation, and it is possible that not all representations of an incoherent channel satisfy Eq. (6) above. The guarantee of incoherent operations are used herein to improve both memory and time costs of the quantum trajectories simulations performed by the comprehensive device simulation system 102. To demonstrate this, classical registers are introduced to the quantum Attorney Docket No. 56113-0504WO1 state vector to track which subspace the state vector is in. A general (possibly coherent) state can be decomposed as ௌ ห^^^^ ൌ ^ ^^^ |^^^^ ⊗ |^^^ ^7^where |^^ ^^ ∈ ℋ^ is a subspace n. Each Krausoperator of an incoherent channel (with Kraus operators satisfying ^^^^^^ ൌ ^^^^^^^^^) cansimilarly be decomposed as ௌ ^^^ఫ ൌ ^ ^^^ೕ^^^^^^^^^ ⊗ ห^^^^^^^^^^^| ^8^ where ^^^^^^^ is the unique index such that ^^^^^^^^^ ് 0. The action of a given Kraus operatoron the state will then be ௌ ⊗^9^ where the classical register still reflects the supporting subspace. Importantly, the memory required to represent a vector |^^^^⊗ |n^ in a single subspace is ^^ ^^^^^^^^ℋ^^^ ^ ^^^^^^^^^^^^^^,which can be exponentially than ^^^^^^^ℋ^ ൌ ^^^^^^^^^ℋ^^. The time required to applythe operation ^^^ೕ^^^^^^|^^^^ can be similarly suppressed, since only the input and output subspaces need to be Therefore, if the initial quantum state has support in a single subspace (i.e. ^^^ ് 0 for exactly one n), the cost of a trajectories simulation will be setby ^^^^^^ ^^^^^^^^ℋ^^^ of ^^^^^^^ℋ^. a quantum channel is not incoherent with respect to the relevant subspaces, the comprehensive device simulation system 102 includes a localized error simulator 108 that includes information about physical error mechanisms in the quantum computing device 110. The localized error simulator 108 is configured to obtain device design schematics 106 for the quantum computing device 110 and convert the design Attorney Docket No. 56113-0504WO1 schematics 106 to Kraus operators to be used in the simulation performed by the comprehensive device simulation system 102. The design schematics 106 can include information about the architecture, components, and interactions in the quantum computing device 110, e.g., information relating to the qubits and couplers included in the quantum computing device, quantum gates and circuits implemented by the quantum computing device, control systems included in the quantum computing device, measurement operations or error correction schemes performed by the quantum computing device. The localized error simulator 108 is configured to apply an approximation referred to herein as a Random Phase Approximation (RPA) to generate an equivalent incoherent channel. The approximation can be used in the context of, e.g., leakage in superconducting qubits where some leakage mechanisms are coherent in nature. The approximation is motivated by the fact that, due to the transmon nonlinearity, the leaked states rapidly accumulate a phase (relative to the computational basis states), e.g., over the course of a single round of error correction. For state |2^ this phase is of the form ^^^^^^^െ^^2^^^^^^^^, where ^^^∼ 1μs is the duration of an error correction round. For typical nonlinearities ∼ 200MHz the phase oscillates rapidly, which suggests that it can be treated as random. To model thisrandom accumulation of phase between subspaces, a phase unitary ^^^^ത^^ ൌ ∑^ ^^^థ^^^^ whichassigns a distinct phase ^^^to each subspace is considered. The RPA is equivalent to twirling over the set of unitary operators This meansconjugating with respect to ^^^^ത^^and its inverse and averaging over independent anduniformly distributed phases ^^^, that is ℰ→ ℰோ^^ ൌ ^^^^ ^െ^ത^^ ◦ ℰ ◦ ^^^ ^^ത^^^థഥ . ^10^Expanding Eq. (10) out gives ^^^^ ^ ^^^^^^^^^^^^^^^^^ற^ ^^^ ^ ^^ ^^ Attorney Docket No. 56113-0504WO1 where the second term corresponds to the case that m=n, m’= n’ and the first to the case that n = n’, m=m’ and not m=n, m’= n’. The Kraus operators generated under the RPA can take two possible forms. The first form includes cross terms ^^^^^^^^^, (12) which can be associated with incoherent transitions. The second form is the block diagonal part of ^^^, Δ^^^^^ ൌ ∑^ ^^^^^^^^^ (13)which preserves each subspace. There are several important properties of the RPA. First, although the Kraus operators generated in Eqs. (12) and (13) depend on the specific choice of Kraus operators ^^^used to represent ℰ, the resulting channel in Eq. (11) is representation independent. Thus, if^^^^^, ^^^^ᇱ^ are two sets of distinct Kraus representations of ℰ, the resulting Kraus operators from equations (12), (13) will correspond to the same channel. Second, observe that bothforms of Kraus operators for RPA satisfy ^^^^^^ ൌ ^^^^^^^^^. Therefore, ℰோ^^ is an incoherentchannel for every ℰ. In fact, ℰோ^^satisfies an even stronger condition: Inserting the Krausoperators of ℰோ^^ into ^^^^^^ ൌ ^^^^^^^^^ shows that for each fixed j, the function ^^^^^,^^^ isinjective in n (i.e., each output has at most one unique input). This classifies ℰோ^^as a strictly incoherent operation. This class of operations is a strict subset of the incoherent operations. Third, it is noted that if one of the projectors represents the full computational subspace, then the RPA preserves error channel process fidelity. The device design module 104 uses results of the simulation performed by the comprehensive device simulation system 102, e.g., simulation statistics, to determine values of metrics that characterize the performance of the quantum computing device, e.g., leakage population or logical error probability of the noisy quantum computation. The device design module 104 can then update the device schematics 106 accordingly. For example, the device design module 104 can determine whether to introduce dedicated error-mitigating operations (e.g., quantum gates that remove leakage) in the quantum computation to improve the Attorney Docket No. 56113-0504WO1 operational efficiency and functionality of the quantum computing device 110. For example, the device design module 104 can determine to introduce dedicated error-mitigating operations in the quantum computation if the leakage population or logical error probability of the noisy quantum computation exceeds a predetermined acceptable threshold. In some implementations the device design module 104 can implement quantum error mitigation techniques or apply leakage detection protocols to identify suitable locations at which to add the dedicated error-mitigating operations. The process can then be iterated to improve the device design, e.g., update the device design to improve operation of the device (e.g., mitigate errors). In implementations that the device design module 104 determines to introduce dedicated error-mitigating operations in the quantum computation, e.g., quantum circuit, device design module 104 can provide the quantum computing device 110 with data representing the updated quantum circuit to the quantum computing device 110 for execution. In response to receiving the data representing the updated quantum circuit, the quantum computing device 110 can execute the updated quantum circuit to perform the quantum computation with reduced leakage population or logical error probability. The quantum computing device 110 can be one of various forms of quantum computing devices. Generally, the quantum computing device 110 can include control electronics that are configured to convert instructions, e.g., digital signals representing control sequences, to control signals required to perform corresponding operations, e.g., quantum gates, on a system of qubits. For example, the control electronics can include control devices that operate physical qubits included in the system of qubits. Example control devices include arbitrary waveform generators or control devices that tune frequencies of respective qubits by applying driving signals, e.g., voltage pulses, to the qubits through respective control lines. The system of qubits includes physical qubits for performing quantum computations. The type of qubits included in the system of qubits is dependent on the types of computations being performed by the quantum computing device 110. For example, in some cases the system can include one or more resonators attached to one or more superconducting qubits, e.g., Gmon or Xmon qubits. In other cases the system can include ion traps, photonic devices or superconducting cavities. Further examples of realizations of qubits include fluxmon qubits, silicon quantum dots or phosphorus impurity qubits. An example process for determining an effect of leakage in a noisy quantum computation (represented by a quantum channel ℰ acting on a subset ^^ of the qubit degrees Attorney Docket No. 56113-0504WO1 of freedom provided by the system of qubits) performed by the example computing system 100 is as follows. The device design module 104 decomposes the local Hilbert space of the subset ^^ as a tensor product ℋൌ ^^∈ ^^ ൫^^^ ⊕ ℒ^൯ ^14^where ^^^represents a local q and ℒ^represents a leakage subspace (this space can be assumed to be one dimensional and spanned by the leakage state |2^, though generalizing to more states is straightforward). The decomposition of the Hilbert space defines projection operators ^^^̅, since the local Hilbert space ℋ can be rewritten as a direct sum ℋൌ⊕^̅ ^^^̅^^^̅ where ^̅^ represents a vector of ^^^ ∈ ^^^, 2^, and ^^^^ ∈^^^^,ℒ^^. The basis for each projection operator ^^^̅can be product of Pauli operators ^^ ^̅ ൌ ^^^భ^^^^మ^..^^^|^^| , ^17^where written as a matrix, the Pauli operator ^^^^has shape 3×2 (for ^^^= c) or 3 × 1 (for ^^^= 2). The localized error simulator 108 applies the RPA to the quantum channel. Under application of the RPA, the Kraus operators ^^^of the quantum channel are transformed to new Kraus operators as in Eqs. (10) and (11). The new operators are either block diagonal with respect to ^^^̅or are off-diagonal with exactly one non-zero block. Each Kraus operator ^^^under the transformation can therefore be mapped to its blocks: ^^^,^̅,௧̅ ൌ ^^ற^̅^^^^^௧̅^18^ Attorney Docket No. 56113-0504WO1 These blocks are used to express ℰோ^^as a combined quantum-classical channel with the following Kraus operators: ^^^^,^̅,௧̅ ⊗ |^̅^^^^^̅|^^,^,̅ஷ௧̅^^^ ^^^,௧̅,௧̅ ⊗ |^^̅^^^^̅|ൡ . ^19^The first type of ^^^̅, while the second represent applying Kraus operator with index j and remaining in a given subspace. For an initial state that is incoherent with respect to the subspaces ^^^̅, the action of ℰோ^^is unchanged if each of the “block-diagonal” Kraus operators is replaced with multiple “singlesubspace” operators ^^^,௧̅,௧̅ ⊗ |^^̅^^^^̅|. This results in a simpler set of Kraus operators, This representation is but more convenient for book- keeping purposes. The structure denoted by Eq. (20) can be stored as a HashMap from pairs^^̅^,ഥ ^^^ to the corresponding blocks ^^^^భ,^̅,௧̅,^^^మ,^̅,௧̅, ... ^. For each “transition” ^^̅ → ^̅^ thesequence runs through only a subset of all j, since only the blocks which are non-zero (up to a predetermined truncation tolerance) are kept. Accordingly, the sequence of the indices^^^^, ^^ଶ, ... ^ is also maintained.The Kraus operator representation in Eq. (20) is not in one of the standard forms described above with reference to Eqs. (12) or (13). (Specifically, the standard form does not allow a Kraus operator to act non-trivially on both existing quantum and classical degrees of freedom.) To map it to the desired form, the representation is separated into two separatechannels. The first channel has Kraus operators of the form ^^^^,^̅,௧̅ ⊗ |^^̅^^^^̅| ⊗|^̅^^^^,^̅,௧̅.Writing the (quantum and classical) state prior to application of |^^̅^, thefirst channel represents the application of the quantum Kraus operators ^^^,^̅,௧̅, conditioned on the current leakage subspace being ^^௧̅. Additionally, it creates a set of classical registers to denote which subspace ^^^̅to which the qubits transitioned. In a trajectories simulation, this corresponds to a lookup of which subspace ^^௧̅the system is currently in, followed by asampling of the Kraus operators ^^^,^̅,௧̅ (for fixed ^^̅ but variable ^^, ^̅^) on the quantum state Attorney Docket No. 56113-0504WO1 vector. After this channel is applied, the “recorded” classical registers (with values ^̅^) are consumed to map the original classical registers to their new values. This is denoted by the second channel (acting only on the classical registers), whose Kraus operators are of the form ^|^̅^^^^^̅| ⊗^^̅^|^ ^̅,௧̅. (This “classical” channel acts trivially on the quantum degrees of freedom.)The simulation described above can be directly generalized in several ways. First, the sampled Kraus operator index j (prior to applying the RPA) can also be recorded in the above decomposition, through the modification |^̅^^ → |^^^ ⊗ |^̅^^ . This can be used to represent ameasurement outcome or capture some other physical process that models an interaction between quantum and classical degrees of freedom. Second, the original Kraus operators need not be square matrices. For example, consider the destructive measurement of a qubit and subsequent recording of the outcome to a classical register, which has Kraus operators ^^^^| ⊗ |^^^^^ . Under the RPA representation, this would correspond to having a single initialsubspace register t and zero output subspace registers. Finally, the RPA is not limited to the case of computational and leakage subspaces. It applies in any situation where the full system Hilbert space can be decomposed as in Eqs. (15) and (16). FIG.2 is a diagram 200 of the dynamics of a quantum-classical state vector in an iteration of a quantum trajectory simulation performed by the comprehensive device simulation system 102 of FIG.1. The Hilbert space 202 for the quantum trajectory simulation is decomposed into multiple Hilbert space subspaces. Each subspace is associated with respective projection operators that define reduced Kraus operators. At the beginning of the iteration, the quantum-classical state vector 204 is written as |^^,^^^, where ^^ represents the quantum degrees of freedom and n represents the classical degrees of freedom. During the iteration, a lookup operation is performed on the classical degrees of freedom to determine that the current Hilbert space subspace is the n-th subspace and Kraus operators ^^^^^^,^for the current Hilbert space subspace are sampled (for fixed n). The sampled Kraus applied to the quantum-classical state vector 206. The classical degrees of freedom are then updated to record the new Hilbert space subspace index m for the next iteration in the quantum trajectories simulation. At the end of the iteration, the quantum- classical state vector 208 is written as |^^^^^^,^^^,^^^. Attorney Docket No. 56113-0504WO1 FIG.3 is a flow diagram of a first example process 300 for determining an effect of uncontrolled quantum state transitions in a noisy quantum computation performed by a quantum computing device, where the noisy quantum computation includes execution of a quantum circuit that is represented by one or more quantum channels (that can include coherent quantum channels). In some implementations the quantum circuit is a quantum error correction circuit. For convenience, the process 300 will be described as being performed by a computing system. For example, the system 100 of FIG.1, appropriately programmed in accordance with this specification, can perform the process 300. The system determines a Hilbert space decomposition for the uncontrolled quantum state transitions in the noisy quantum computation (step 302). In some implementations the uncontrolled quantum state transitions include transitions of a single qudit out of the computational subspace and into a leakage subspace. In these implementations, the Hilbert space can be decomposed into a first subspace (a “computational subspace) spanned by computational quantum states 0 and 1 and a second subspace (a “leakage subspace”) spanned by a leakage quantum state 2, e.g., ℋൌ ^^ ⊕ ℒ^^ ൌ ^^^^^^^^^|0^, |1^^ℒ ൌ ^^^^^^^^^|2^^ .where the states |0^, |1^, |2^ represent the lowest energy eigenstates of the Hamiltonian, e.g., Transmon Hamiltonian. It is said that leakage in the system is incoherent if the system state only ever has support in either ^^ or ℒ. Using the construction above, this means the memory requirements for the simulation at step 306 below are ^^^2^^ for ^^ such qudits instead of 3^. In other implementations the uncontrolled quantum state transitions can include transitions of n qudits out of the computational subspace and into the leakage subspace. In these implementations the Hilbert space can be decomposed into 2nsubspaces (a computational subspace and a leakage subspace for each qudit). In other implementations the uncontrolled quantum state transitions can include coupler state transitions or transitions of a two-level system (TLS). In these implementations the Hilbert space can be decomposed into a first subspace that corresponds to the coupler or Attorney Docket No. 56113-0504WO1 TLS being in a zero state and a second subspace that corresponds to the coupler or TLS being in a one state. The system determines projection operators for each subspace in the Hilbert space decomposition (step 304). The projection operators define reduced Kraus operators for the simulation, as described above with reference to Eqs. (12) and (13). The system performs a simulation of the quantum circuit represented by the one or more quantum channels using a quantum-classical state of a qubit system and a classical register (step 306). In implementations where the one or more quantum channels that represent the quantum circuit are not incoherent, the system can apply the random phase approximation to the quantum channels before performing the simulation at step 306, as described below with reference to FIG.4. The simulation can be a discrete time quantum trajectory simulation. To perform a discrete time quantum trajectory simulation, for each quantum channel in the composition of quantum channels that represents the quantum circuit, the system iteratively evolves the quantum-classical state from an initial quantum state (e.g., a pure state) to a final state, where a final state obtained for a preceding incoherent quantum channel in the composition of quantum channels is provided as an initial state for a subsequent quantum channel in the composition. At each iteration, the system performs a classical computation to identify a current Hilbert space subspace occupied by the quantum state for the iteration by performing a lookup in the classical registers that each record a Hilbert space subspace occupied by a respective qubit in the system of qubits (step 308). The system then performs a classical computation to sample terms of reduced Kraus operators that act on the current Hilbert space subspace (step 310). The system then applies the sampled terms of the reduced Kraus operators to the quantum degrees of freedom of the state for the iteration (step 312). The system then updates the classical registers based on the sampled terms of the reduced Kraus operators (314). The simulation only simulates a single subspace at a time, which limits the amount of required memory. The system processes results of the simulation to obtain simulation statistics that represent the effect of the uncontrolled quantum state transitions in the noisy quantum computation performed by the quantum computing device (step 316). For example, each trajectory in the simulation can end with a measurement of the final state. The system can compare the measurement results obtained from the different trajectories to an expected Attorney Docket No. 56113-0504WO1 outcome of the quantum computation in an absence of noise and then determine, based on the comparison, a leakage population or logical error probability of the noisy quantum computation. The system can determine, based on the leakage population or logical error probability of the noisy quantum computation, whether to introduce dedicated error- mitigating operations, such as quantum gates that remove leakage, in the quantum computation. In other words, the quantum computation can be calibrated to reduce the leakage population or logical error probability by introducing dedicated error-mitigating operations. FIG.4 is a flow diagram of a second example process 400 for determining an effect of uncontrolled quantum state transitions in a noisy quantum computation performed by a quantum computing device, where the noisy quantum computation includes execution of a quantum circuit that is represented by one or more quantum channels that are not incoherent. In some implementations the quantum circuit is a quantum error correction circuit. For convenience, the process 400 will be described as being performed by a computing system. For example, the system 100 of FIG.1, appropriately programmed in accordance with this specification, can perform the process 400. The system approximates the uncontrolled quantum state transitions as incoherent uncontrolled quantum state transitions. This includes, applying the random phase approximation described herein to quantum channels that represent the quantum circuit (step 402). The random phase approximation imposes incoherence between computational subspaces and leakage subspaces for the one or more quantum channels. To apply the random phase approximation to a quantum channel, the system conjugates the quantum channel with respect to a phase unitary operator, where the phase unitary operator is an operator that assigns a distinct random phase to each subspace in a decomposition of a Hilbert space for the noisy quantum circuit into a direct sum of subspaces. The system then conjugates the quantum channel with respect to an inverse of the phase unitary operator, and averages over a uniform distribution of the distinct random phases. See, e.g., Eq. (10) above. The quantum channels that represent the quantum circuit are represented by respective Kraus operators that act on a full Hilbert space for the simulation, as described above with reference to FIG.1 (see, e.g., Eq. (1). Application of the random phase approximation to the quantum channels generates reduced Kraus operators, where the reduced Kraus operators act on subspaces of the full Hilbert space. The reduced Kraus Attorney Docket No. 56113-0504WO1 operators include projection operators onto respective subspaces of the full Hilbert space, wherein the number of subspaces is dependent on the type of uncontrolled quantum state transitions being simulated. The reduced Kraus operators include off-diagonal terms associated with incoherent transitions, where each off-diagonal term comprises a projection operator onto a first subspace of the full Hilbert space, a Kraus operator that acts on the full Hilbert space, and a projection operator onto a second subspace of the Hilbert space, wherein the second subspace is different from the first subspace. The reduced Kraus operators also include block diagonal terms that preserve Hilbert space subspaces, where each block diagonal term comprises a sum, over one or more subspaces of the Hilbert space, of operators comprising a projection operator onto the subspace, a Kraus operator that acts on the full Hilbert space, and the projection operator onto the subspace. See, e.g., Eq. (12) and (13) above. Application of the random phase approximation to the quantum channels produces a modified quantum circuit that is represented by incoherent quantum channels. The system then performs a simulation of the modified quantum circuit to simulate the (originally coherent) uncontrolled quantum state transitions (step 404). By construction, the simulation of the modified quantum circuit only requires a simulation of incoherent leakage, which means that the simulation can be performed using qubit resources only (i.e., the simulation is a qubit simulation). In some implementations the simulation of the modified quantum circuit can be performed using a discrete time quantum trajectory simulation, e.g., using a quantum- classical state of a qubit system and a classical register that tracks the Hilbert space subspace of the state of the qubit system, as described above with reference to steps 302-306 of example process 300 of FIG.3. The system processes results of the simulation to obtain simulation statistics that represent the effect of the uncontrolled quantum state transitions in the noisy quantum computation performed by the quantum computing device, e.g., leakage populations or logical error rates for the noisy quantum computation (step 406). Step 406 of example process 400 is similar to step 316 of example process 300. For brevity, details are not repeated. Implementations of the subject matter and operations described in this specification can be implemented in digital electronic circuitry, analog electronic circuitry, suitable quantum circuitry or, more generally, quantum computational systems, in tangibly-embodied software or firmware, in computer hardware, including the structures disclosed in this Attorney Docket No. 56113-0504WO1 specification and their structural equivalents, or in combinations of one or more of them. The term “quantum computational systems” may include, but is not limited to, quantum computers, quantum information processing systems, quantum cryptography systems, or quantum simulators. Implementations of the subject matter described in this specification can be implemented as one or more computer programs, i.e., one or more modules of computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, data processing apparatus. The computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of them. Alternatively or in addition, the program instructions can be encoded on an artificially- generated propagated signal that is capable of encoding digital and / or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal, that is generated to encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus. The terms quantum information and quantum data refer to information or data that is carried by, held or stored in quantum systems, where the smallest non-trivial system is a qubit, i.e., a system that defines the unit of quantum information. It is understood that the term “qubit” encompasses all quantum systems that may be suitably approximated as a two- level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions or superconducting qubits. In many implementations the computational basis states are identified with the ground and first excited states, however it is understood that other setups where the computational states are identified with higher level excited states are possible. The term “data processing apparatus” refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus that is designed to simulate or produce information about Attorney Docket No. 56113-0504WO1 a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. The apparatus can optionally include, in addition to hardware, code that creates an execution environment for digital and / or quantum computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them. A digital computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or can be written in a quantum programming language, e.g., QCL or Quipper. A computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, e.g., files that store one or more modules, sub- programs, or portions of code. A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that may transmit quantum data using quantum systems, e.g. qubits. Generally, a digital data communication network cannot transmit quantum data, however a quantum data communication network may transmit both quantum data and digital data. The processes and logic flows described in this specification can be performed by one or more programmable computers, operating with one or more processors, as appropriate, executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA or an Attorney Docket No. 56113-0504WO1 ASIC, or a quantum simulator, or by a combination of special purpose logic circuitry or quantum simulators and one or more programmed digital and / or quantum computers. For a system of one or more computers to be “configured to” perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by data processing apparatus, cause the apparatus to perform the operations or actions. For example, a quantum computer may receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operations or actions. Computers suitable for the execution of a computer program can be based on general or special purpose processors, or any other kind of central processing unit. Generally, a central processing unit will receive instructions and data from a read-only memory, a random access memory, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof . The elements of a computer include a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital, analog, and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special purpose logic circuitry or quantum simulators. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, a computer need not have such devices. Quantum circuit elements (also referred to as quantum computing circuit elements) include physical circuit elements for performing quantum processing operations. That is, the quantum circuit elements are configured to make use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data in a non-deterministic manner. Certain quantum circuit elements, such as qubits, can be configured to represent and operate on information in more than one state simultaneously. Examples of superconducting quantum circuit elements include circuit elements such as quantum LC oscillators, qubits (e.g., flux qubits, phase qubits, or charge qubits), and superconducting quantum interference devices (SQUIDs) (e.g., RF-SQUID or DC-SQUID), among others. Attorney Docket No. 56113-0504WO1 In contrast, classical circuit elements generally process data in a deterministic manner. Classical circuit elements can be configured to collectively carry out instructions of a computer program by performing basic arithmetical, logical, and / or input / output operations on data, in which the data is represented in analog or digital form. In some implementations, classical circuit elements can be used to transmit data to and / or receive data from the quantum circuit elements through electrical or electromagnetic connections. Examples of classical circuit elements include circuit elements based on CMOS circuitry, rapid single flux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices and ERSFQ devices, which are an energy-efficient version of RSFQ that does not use bias resistors. In certain cases, some or all of the quantum and / or classical circuit elements may be implemented using, e.g., superconducting quantum and / or classical circuit elements. Fabrication of the superconducting circuit elements can entail the deposition of one or more materials, such as superconductors, dielectrics and / or metals. Depending on the selected material, these materials can be deposited using deposition processes such as chemical vapor deposition, physical vapor deposition (e.g., evaporation or sputtering), or epitaxial techniques, among other deposition processes. Processes for fabricating circuit elements described herein can entail the removal of one or more materials from a device during fabrication. Depending on the material to be removed, the removal process can include, e.g., wet etching techniques, dry etching techniques, or lift-off processes. The materials forming the circuit elements described herein can be patterned using known lithographic techniques (e.g., photolithography or e-beam lithography). During operation of a quantum computational system that uses superconducting quantum circuit elements and / or superconducting classical circuit elements, such as the circuit elements described herein, the superconducting circuit elements are cooled down within a cryostat to temperatures that allow a superconductor material to exhibit superconducting properties. A superconductor (alternatively superconducting) material can be understood as material that exhibits superconducting properties at or below a superconducting critical temperature. Examples of superconducting material include aluminum (superconductive critical temperature of 1.2 kelvin) and niobium (superconducting critical temperature of 9.3 kelvin). Accordingly, superconducting structures, such as superconducting traces and superconducting ground planes, are formed from material that exhibits superconducting properties at or below a superconducting critical temperature. Attorney Docket No. 56113-0504WO1 In certain implementations, control signals for the quantum circuit elements (e.g., qubits and qubit couplers) may be provided using classical circuit elements that are electrically and / or electromagnetically coupled to the quantum circuit elements. The control signals may be provided in digital and / or analog form. Computer-readable media suitable for storing computer program instructions and data include all forms of non-volatile digital and / or quantum memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto- optical disks; CD-ROM and DVD-ROM disks; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence. Control of the various systems described in this specification, or portions of them, can be implemented in a computer program product that includes instructions that are stored on one or more non-transitory machine-readable storage media, and that are executable on one or more processing devices. The systems described in this specification, or portions of them, can each be implemented as an apparatus, method, or system that may include one or more processing devices and memory to store executable instructions to perform the operations described in this specification. While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub-combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination. Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular Attorney Docket No. 56113-0504WO1 order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the implementations described above should not be understood as requiring such separation in all implementations, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products. Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous. What is claimed is:

Claims

Attorney Docket No. 56113-0504WO1 CLAIMS 1. A method performed by a quantum computing device, the method comprising: determining an effect of uncontrolled quantum state transitions in a noisy quantum computation performed by the quantum computing device, wherein the noisy quantum computation comprises execution of a quantum circuit that is represented by one or more quantum channels, the determining comprising: approximating the uncontrolled quantum state transitions as incoherent uncontrolled quantum state transitions, comprising applying a random phase approximation to the one or more quantum channels to obtain a modified quantum circuit that is represented by one or more incoherent quantum channels, wherein the incoherent quantum channels preserve incoherence between computational subspaces and environmental subspaces for the one or more quantum channels; and performing a simulation of the modified quantum circuit, comprising performing a qubit simulation of the modified quantum circuit.

2. The method of claim 1, wherein the simulation comprises a discrete time quantum trajectory simulation.

3. The method of claim 2, wherein performing the discrete time quantum trajectory simulation comprises, for each incoherent quantum channel: iteratively evolving a quantum-classical state of a system of qubits and a classical register included in the quantum computing device from an initial quantum-classical state to a final quantum-classical state, wherein the classical register records Hilbert space subspaces occupied by the qubits in the system of qubits during the evolving.

4. The method of claim 3, wherein the modified quantum circuit is represented by a composition of the incoherent quantum channels and wherein a final quantum-classical state obtained for a preceding incoherent quantum channel in the composition of incoherent quantum channels is provided as an initial quantum-classical state for a subsequent incoherent quantum channel in the composition.

5. The method of claim 3, wherein iteratively evolving comprises, for each iteration:Attorney Docket No. 56113-0504WO1 identifying, by classical computation, a current Hilbert space subspace occupied by quantum degrees of freedom of the quantum-classical state for the iteration, comprising performing a lookup in the classical register; sampling, by classical computation, terms of reduced Kraus operators that act on the current Hilbert space subspace; applying, by quantum computation, the sampled terms of the reduced Kraus operators to the quantum degrees of freedom of the state for the iteration; and updating, by classical computation, the classical register based on the sampled terms of the reduced Kraus operators.

6. The method of any one of claims 1 to 5, wherein the random phase approximation comprises an average over a uniform distribution of distinct random phases of: a conjugate of the quantum channel with respect to a phase unitary operator, wherein the phase unitary operator comprises an operator that assigns a distinct random phase to each subspace in a decomposition of a Hilbert space for the quantum circuit into a direct sum of subspaces; and a conjugate of the quantum channel with respect to an inverse of the phase unitary operator.

7. The method of any one of claims 1 to 6, wherein: the one or more quantum channels are represented by respective Kraus operators that act on a full Hilbert space for the simulation; and applying the random phase approximation to the one or more quantum channels comprises generating reduced Kraus operators that act on subspaces of the full Hilbert space.

8. The method of claim 7, wherein the reduced Kraus operators comprise off-diagonal terms associated with incoherent transitions, wherein each off-diagonal term comprises a projection operator onto a first subspace of the full Hilbert space, a Kraus operator that acts on the full Hilbert space, and a projection operator onto a second subspace of the Hilbert space, wherein the second subspace is different from the first subspace.

9. The method of claim 7, wherein the reduced Kraus operators comprise block diagonal terms that preserve Hilbert space subspaces, wherein each block diagonal term comprises aAttorney Docket No. 56113-0504WO1 sum, over one or more subspaces of the Hilbert space, of operators comprising a projection operator onto the subspace, a Kraus operator that acts on the full Hilbert space, and the projection operator onto the subspace.

10. The method of claim 7, wherein the reduced Kraus operators comprise projection operators onto respective subspaces of the full Hilbert space, wherein the number of subspaces is dependent on a type of uncontrolled quantum state transitions.

11. The method of any preceding claim, further comprising: determining a Hilbert space decomposition for the uncontrolled quantum state transitions in the noisy quantum computation; and determining projection operators for each subspace in the Hilbert space decomposition, wherein the projection operators define reduced Kraus operators for the simulation.

12. The method of claim 11, wherein uncontrolled quantum state transitions comprise transitions of a single qudit out of the computational subspace and into a leakage subspace, and wherein the Hilbert space decomposition comprises a first subspace spanned by computational quantum states 0 and 1, and a second subspace spanned by a leakage quantum state 2.

13. The method of claim 11, wherein the uncontrolled quantum state transitions comprise transitions of n qudits out of the computational subspace and into a leakage subspace, wherein the Hilbert space decomposition comprises 2nsubspaces.

14. The method of claim 11, wherein the uncontrolled quantum state transitions comprise coupler state transitions or transitions of a two-level system (TLS), and wherein the subspaces of the Hilbert space comprise a first subspace corresponding the coupler or TLS being in a zero state and a second subspace corresponding to the coupler or TLS being in a one state.

15. The method of any preceding claim, further comprising processing results of the simulation to obtain simulation statistics that represent the effect of the uncontrolled quantumAttorney Docket No. 56113-0504WO1 state transitions in the noisy quantum computation performed by the quantum computing device.

16. The method of claim 15, wherein processing the results comprises: comparing the results to an expected outcome of the quantum computation in an absence of noise; and determining, based on the comparison, a leakage population or logical error probability of the noisy quantum computation.

17. The method of any one of claims 1 to 16, wherein performing the qubit simulation of the modified quantum circuit comprises simulating single Hilbert space subspaces.

18. The method of any one of claims 1 to 17, wherein the quantum circuit comprises a quantum error correction circuit.

19. A system comprising one or more computers and one or more storage devices storing instructions that are operable, when executed by the one or more computers, to cause the one or more computers to perform operations comprising the method of any preceding claim.

20. One or more computer storage media encoded with instructions that, when executed by one or more computers, cause the one or more computers to perform operations comprising the method of any preceding claim.

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